297
Source: NASA.
5.1 Rules for Exponents 5.2 Addition and Subtrac-
tion of Polynomials 5.3 Multiplication of
Polynomials 5.4 Special Products 5.5 Integer Exponents and
the Quotient Rule 5.6 Division of Polynomials
Digital images were first sent between New York and London by cable in the early 1920s. Unfortunately, the transmission time was 3 hours and the quality was poor. Digital photography was developed further by NASA in the 1960s because ordinary pic- tures were subject to interference when transmitted through space. Today, digital pic- tures remain crystal clear even if they travel millions of miles. The following digital picture shows the planet Mars.
Whether they are taken with a webcam, with a smartphone, or by the Mars rover, digital images comprise tiny units called pixels, which are represented by numbers. As a result, mathematics plays an important role in digital images. In this chapter we illustrate some of the ways mathematics is used to describe digital pictures (see Example 4 and Exercise 80 in Section 5.4). We also discuss how mathematics is used to model things such as heart rate, computer sales, motion of the planets, and interest on money.
5 Polynomials and Exponents
If you want to do something, do it!
—PLAUTUS
IS B
N 1-
25 6-
49 08
2- 2
Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.
298 CHAPTER 5 POLYNOMIALS AND EXPONENTS
When evaluating expressions, evaluate exponents before performing addition, subtraction, multiplication, division, or negation.
5.1 Rules for Exponents Review of Bases and Exponents ● Zero Exponents ● The Product Rule ● Power Rules
A LOOK INTO MATH N Electronic devices such as tablet computers and smartphones store information as bits. A bit is either a 0 or a 1, and a string of 8 bits is called a byte. In the 1970s, IBM devel- oped punch cards made out of paper that could hold up to 120 bits of information. Today, many computer hard drives can hold more than 1 terabyte of information; that’s more than 8,000,000,000,000 bits! In mathematics, we often use exponents to express such large numbers. In this section, we discuss the rules for exponents.
Review of Bases and Exponents The expression 53 is an exponential expression with base 5 and exponent 3. Its value is
5 # 5 # 5 = 125. In general, b n is an exponential expression with base b and exponent n. If n is a natural number, it indicates the number of times the base b is to be multiplied with itself.
Exponent T
b n = b # b # b # g # b Base c n times
v
STUDY TIP
Exponents occur throughout mathematics. Because expo- nents are so important, this section is essential for your success in mathematics. It takes practice, so set aside some extra time.
EVALUATING EXPRESSIONS
When evaluating expressions, use the following order of operations.
1. Evaluate exponents. 2. Perform negation. 3. Do multiplication and division from left to right. 4. Do addition and subtraction from left to right.
EXAMPLE 1 Evaluating exponential expressions
Evaluate each expression.
(a) 1 + 24
4 (b) 3a1
3 b2 (c) - 24 (d) ( - 2)4
Solution (a) Evaluate the exponent first.
4 factors
1 + 24
4 = 1 +
2 # 2 # 2 # 2 4
= 1 + 16
4 = 1 + 4 = 5
∂
2 factors
(b) 3a1 3 b2 = 3a1
3 # 1
3 b = 3 # 1
9 =
3
9 =
1
3
f
IS B
N 1-256-49082-2
Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.
2995.1 RULES FOR EXPONENTS
(c) Because exponents are evaluated before negation is performed,
4 factors
- 24 = - (2 # 2 # 2 # 2) = - 16.∂ 4 factors
(d) (�2)4 = (�2)(�2)(�2)(�2) = 16
v
Now Try Exercises 9, 11, 15, 17
NOTE: Parts (c) and (d) of Example 1 appear to be very similar. However, the negation sign is inside the parentheses in part (d), which means that the base for the exponential expression is - 2. In part (c), no parentheses are used, indicating that the base of the expo- nential expression is 2.
READING CHECK
• Explain how to tell the difference between a negative number raised to a power and the opposite of a positive number raised to a power.
TECHNOLOGY NOTE
Evaluating Exponents Exponents can often be evaluated on calculators by using the ^ key. The four expressions from Example 1 are evaluated with a calculator and the results are shown in the following two figures. When evaluating the last two expressions on your calculator, remember to use the negation key rather than the subtraction key.
1�2^4/4 5
3(1/3) 2�Frac 1/3
�2^4 �16
(�2)^4 16
CALCULATOR HELP To evaluate exponents, see Appendix A (page AP-1).
Zero Exponents So far we have discussed natural number exponents. What if an exponent is 0? What does 20 equal? To answer these questions, consider Table 5.1, which shows values for decreas- ing powers of 2. Note that each time the power of 2 decreases by 1, the resulting value is divided by 2. For this pattern to continue, we need to define 20 to be 1 because dividing 2 by 2 results in 1.
This discussion suggests that 20 = 1, and is generalized as follows.
TABLE 5.1 Powers of 2
Power of 2 Value
23 8
22 4
21 2
20 ? ZERO EXPONENT
For any nonzero real number b,
b 0 = 1.
The expression 00 is undefined.
IS B
N 1-
25 6-
49 08
2- 2
Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.
300 CHAPTER 5 POLYNOMIALS AND EXPONENTS
EXAMPLE 2 Evaluating zero exponents
Evaluate each expression. Assume that all variables represent nonzero numbers.
(a) 70 (b) 3a4 9 b0 (c) ax 2y5
3z b0
Solution (a) 70 = 1 (b) 314920 = 3(1) = 3. (Note that the exponent 0 does not apply to 3.) (c) All variables are nonzero, so the expression inside the parentheses is also nonzero.
Thus 1x 2y53z 20 = 1. Now Try Exercises 13, 41, 67
THE PRODUCT RULE
For any real number a and natural numbers m and n,
am # an = am + n.
READING CHECK
• State the product rule in your own words.
The expression 43 # 42 has 3 + 2 = 5 factors of 4, so the result is 43 + 2 = 45. To multiply exponential expressions with the same base, add exponents and keep the base.
5 factors
The Product Rule We can use a special rule to calculate products of exponential expressions provided their bases are the same. For example,
43 # 42 = (4 # 4 # 4) # (4 # 4) = 45.v
NOTE: The product 24 # 35 cannot be simplified by using the product rule because the exponential expressions have different bases: 2 and 3.
EXAMPLE 3 Using the product rule
Multiply and simplify. (a) 23 # 22 (b) x 4x 5 (c) 2 x 2 # 5x 6 (d) x 3(2 x + 3x 2) Solution (a) 23 # 22 = 23 + 2 = 25 = 32 (b) x 4x 5 = x 4 + 5 = x 9 (c) Begin by applying the commutative property of multiplication to write the product in a
more convenient order.
2 x 2 # 5x 6 = 2 # 5 # x 2 # x 6 = 10x 2 + 6 = 10x 8 (d) To simplify this expression, first apply the distributive property.
3�1 T T 3�2
x 3(2 x + 3x 2) = x 3 # 2 x + x 3 # 3x 2 = 2 x 4 + 3x 5 c Exponent is 1.
Now Try Exercises 21, 23, 27, 71
IS B
N 1-256-49082-2
Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.
3015.1 RULES FOR EXPONENTS
EXAMPLE 4 Applying the product rule
Multiply and simplify. (a) x # x 3 (b) (a + b)(a + b)4
Solution (a) Begin by writing x as x 1. Then x 1 # x 3 = x 1 + 3 = x 4. (b) First write (a + b) as (a + b)1. Then
(a � b)1 # (a � b)4 = (a � b)1 + 4 = (a � b)5. Now Try Exercises 19, 63
Power Rules How should (43)2 be evaluated? To answer this question, consider how the product rule can be used in evaluating
(43)2 = 43 # 43 = 43 + 3 = 46.
5 Product rule
5 3 + 3 = 3 # 2 Similarly,
5 + 5 + 5 = 5 # 3
$%& (a5)3 = a5 # a5 # a5 = a5 + 5 + 5 = a15.
Product rule
w
This discussion suggests that to raise a power to a power, we multiply the exponents.
RAISING A POWER TO A POWER
For any real number a and natural numbers m and n,
(am)n = amn.
EXAMPLE 5 Raising a power to a power
Simplify each expression. (a) (32)4 (b) (a3)2
Solution (a) (32)4 = 32
# 4 = 38 (b) (a3)2 = a3 # 2 = a6
Now Try Exercises 31, 33
To decide how to simplify the expression (2 x)3, consider
(2 x)3 = 2 x # 2 x # 2 x = (2 # 2 # 2) # (x # x # x) = 23x 3. 3 factors 3 factors 3 factors
∂ ¶ ¶
To raise a product to a power, we raise each factor to the power.
NOTE: If an exponent does not appear in an expression, it is assumed to be 1. For example, x can be written as x 1 and (x + y) can be written as (x + y)1.
IS B
N 1-
25 6-
49 08
2- 2
Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.
302 CHAPTER 5 POLYNOMIALS AND EXPONENTS
READING CHECK
• State the rule for raising a product to a power in your own words.
RAISING A PRODUCT TO A POWER
For any real numbers a and b and natural number n,
(ab)n = anb n.
EXAMPLE 6 Raising a product to a power
Simplify each expression. (a) (3z)2 (b) ( - 2 x 2)3 (c) 4(x 2y3)5 (d) ( - 22a5)3
Solution (a) (3z)2 = 32z2 = 9z2 (b) ( - 2 x 2)3 = ( - 2)3(x 2)3 = - 8x 6 (c) 4(x 2y3)5 = 4(x 2)5( y3)5 = 4 x 10y15 (d) ( - 22a5)3 = ( - 4a5)3 = ( - 4)3(a5)3 = - 64a15
Now Try Exercises 37, 39, 43, 45
The following equation illustrates another power rule.
a2 3 b4 = 2
3 # 2
3 # 2
3 # 2
3 =
2 # 2 # 2 # 2 3 # 3 # 3 # 3 =
24
34
4 factors
t
To raise a quotient to a power, raise both the numerator and the denominator to the power.
RAISING A QUOTIENT TO A POWER
For any real numbers a and b and natural number n,
aa b bn = an
b n . b � 0
EXAMPLE 7 Raising a quotient to a power
Simplify each expression.
(a) a2 3 b3 (b) aa
b b9 (c) aa + b
5 b2
Solution
(a) a2 3 b3 = 23
33 =
8
27 (b) aa
b b9 = a9
b 9
(c) Because the numerator is an expression with more than one term, we must place paren- theses around it before raising it to the power 2.
aa + b 5 b2 = (a + b)2
52 =
(a + b)2
25
Now Try Exercises 51, 53, 55
IS B
N 1-256-49082-2
Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.
3035.1 RULES FOR EXPONENTS
The five rules for exponents discussed in this section are summarized as follows.
MAKING CONNECTIONS
Raising a Sum or Difference to a Power
Although there are power rules for products and quotients, there are not similar rules for sums and differences. In general, (a + b)n 3 an + b n and (a - b)n 3 an - b n. For example, (3 + 4)2 = 72 = 49 but 32 + 42 = 9 + 16 = 25. Similarly, (4 - 1)3 = 33 = 27 but 43 - 13 = 64 - 1 = 63.
RULES FOR EXPONENTS
The following rules hold for real numbers a and b, and natural numbers m and n.
Description Rule Example
Zero Exponent b 0 = 1, for b � 0 ( - 13)0 = 1
The Product Rule am # an = am + n 54 # 53 = 54 + 3 = 57 Power to a Power (am)n = am # n ( y2)5 = y2 # 5 = y10
Product to a Power (ab)n = anb n ( pq)7 = p7q 7
Quotient to a Power aa b bn = an
b n , for b � 0 ax
y b3 = x 3
y3 , for y � 0
Simplification of some expressions may require the application of more than one rule of exponents. This is demonstrated in the next example.
EXAMPLE 8 Combining rules for exponents
Simplify each expression.
(a) (2a)2(3a)3 (b) aa2b 3 c b4 (c) (2 x 3y)2( - 4 x 2y3)3
Solution (a) (2a)2(3a)3 = 22a2 # 33a3 Raising a product to a power = 4 # 27 # a2 # a3 Evaluate powers; commutative property = 108a5 Product rule
(b) aa2b 3 c b4 = (a2)4(b 3)4
c 4 Raising a quotient to a power; raising
a product to a power
= a8b 12
c 4 Raising a power to a power
(c) (2 x 3y)2( - 4 x 2y3)3 = 22(x 3)2y2( - 4)3(x 2)3( y3)3 Raising a product to a power = 4 x 6y2( - 64)x 6y9 Raising a power to a power = 4( - 64)x 6x 6y2y9 Commutative property = - 256 x 12y11 Product rule
Now Try Exercises 47, 49, 61
IS B
N 1-
25 6-
49 08
2- 2
Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.
304 CHAPTER 5 POLYNOMIALS AND EXPONENTS
N REAL-WORLD CONNECTION Exponents occur frequently in calculations involving yearly percent increases, such as the increase in property value illustrated in the next example.
EXAMPLE 9 Calculating growth in property value
If a parcel of property increases in value by about 11% each year for 20 years, then its value will double three times. (a) Write an exponential expression that represents “doubling three times.” (b) If the property is initially worth $25,000, how much will it be worth after it doubles
3 times?
Solution (a) Doubling three times is represented by 23. (b) 23(25,000) = 8(25,000) = $200,000
Now Try Exercise 85
5.1 Putting It All Together
Bases and Exponents In the expression b n, b is the base and n is the exponent. If n is a natural number, then
b n = b # b # g # b. 5 n times
23 has base 2 and exponent 3.
91 = 9, 32 = 3 # 3 = 9, 43 = 4 # 4 # 4 = 64, and - 62 = - (6 # 6) = - 36
Zero Exponents For any nonzero number b, b 0 = 1. 50 = 1, x 0 = 1, and (xy3)0 = 1
The Product Rule For any real number a and natural numbers m and n,
am # an = am + n.
24 # 23 = 24 + 3 = 27, x # x 2 # x 6 = x 1 + 2 + 6 = x 9, and (x + 1) # (x + 1)2 = (x + 1)3
CONCEPT EXPLANATION EXAMPLES
Raising a Power to a Power
For any real number a and natural numbers m and n,
(am)n = amn.
(24)2 = 24 # 2 = 28,
(x 2)5 = x 2 # 5 = x 10, and
(a4)3 = a4 # 3 = a12
Raising a Product to a Power
For any real numbers a and b and natural number n,
(ab)n = anb n.
(3x)3 = 33x 3 = 27x 3, (x 2y)4 = (x 2)4y4 = x 8y4, and ( - xy)6 = ( - x)6y6 = x 6y6
Raising a Quotient to a Power
For any real numbers a and b and natural number n,
aa b bn = an
b n . b � 0
ax y b5 = x 5
y5 and
aa2b d 3 b4 = (a2)4b 4
(d 3)4 =
a8b 4
d 12
IS B
N 1-256-49082-2
Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.
3055.1 RULES FOR EXPONENTS
5.1 Exercises
CONCEPTS AND VOCABULARY
1. In the expression b n, b is the _____ and n is the _____.
2. The expression b 0 = for any nonzero number b.
3. am # an = 4. (am)n = 5. (ab)n = 6. aa
b bn =
PROPERTIES OF EXPONENTS
Exercises 7–18: Evaluate the expression.
7. 82 8. 43
9. ( - 2)3 10. ( - 3)4
11. - 23 12. - 34
13. 60 14. ( - 0.5)0
15. 3 + 42
2 16. 6 - a - 4
2 b2
17. 4a1 2 b3 18. 16a1
4 b2
Exercises 19–74: Simplify the expression. Assume that all variables represent nonzero numbers.
19. 3 # 32 20. 53 # 53 21. 42 # 46 22. 104 # 103
23. x 3x 6 24. a5a2
25. x 2x 2x 2 26. y7y3y0
27. 4 x 2 # 5x 5 28. - 2y6 # 5y2 29. 3( - xy3)(x 2y) 30. (a2b 3)( - ab 2)
31. (23)2 32. (103)4
33. (n3)4 34. (z7)3
35. x(x 3)2 36. (z3)2(5z5)
37. ( - 7b)2 38. ( - 4z)3
39. (ab)3 40. (xy)8
41. (2 x 2)0 42. (3a2)4
43. ( - 4b 2)3 44. ( - 3r 4t 3)2
45. (x 2y3)7 46. (rt 2)5
47. ( y3)2(x 4y)3 48. (ab 3)2(ab)3
49. (a2b)2(a2b 2)3 50. (x 3y)(x 2y4)2
51. a1 3 b3 52. a5
2 b2
53. aa b b5 54. ax
2 b4
55. ax - y 3 b3 56. a 4
x + y b2
57. a 5 a + b
b2 58. aa - b 2 b3
59. a2 x 5 b3 60. a3y
2 b4
61. a3x 2 5y4 b3 62. aa2b 3
3 b5
63. (x + y)(x + y)3 64. (a - b)2(a - b)
65. (a + b)2(a + b)3 66. (x - y)5(x - y)4
67. 6(x 4y6)0 68. axy z2 b0
69. a(a2 + 2b 2) 70. x 3(3x - 5y4) 71. 3a3(4a2 + 2b) 72. 2 x 2(5 - 4y3)
73. (r + t)(rt)
74. (x - y)(x 2y3)
75. Thinking Generally Students sometimes mistakenly apply the “rule” am # b n � (ab)m + n. In general, this equation is not true. Find values for a, b, m, and n with a � b and m � n that will make this equation true.
76. Thinking Generally Students sometimes mistakenly
apply the “rule” (a + b)n � an + b n. In general, this equation is not true. Find values for a, b, and n with a � b that will make this equation true.
APPLICATIONS
Exercises 77–80: Write a simplified expression for the area of the given figure.
77. 2x2
5x2
78. 2ab
2ab
79.
3x2
80.
4y2
7y3
IS B
N 1-
25 6-
49 08
2- 2
Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.
306 CHAPTER 5 POLYNOMIALS AND EXPONENTS
Exercises 81 and 82: Write a simplified expression for the volume of the given figure.
81.
x
2x
4x
82.
3a2
3a2 3a2
83. Compound Interest If P dollars are deposited in an account that pays 5% annual interest, then the amount of money in the account after 3 years is P(1 + 0.05)3. Find the amount when P = $1000.
84. Compound Interest If P dollars are deposited in an account that pays 9% annual interest, then the amount of money in the account after 4 years is P(1 + 0.09)4. Find the amount when P = $500.
85. Investment Growth If an investment increases in value by about 10% each year for 22 years, then its value will triple two times.
(a) Write an exponential expression that represents “tripling two times.”
(b) If the investment has an initial value of $8000, how much will it be worth if it triples two times?
86. Stock Value If a stock decreases in value by about 23% each year for 9 years, then its value will be halved three times. (a) Write an exponential expression that represents
“halved three times.” (b) If the stock is initially worth $88 per share, how
much will it be worth if it is halved three times?
WRITING ABOUT MATHEMATICS
87. Are the expressions (4 x)2 and 4 x 2 equal in value? Explain your answer.
88. Are the expressions 33 # 23 and 66 equal in value? Explain your answer.
5.2 Addition and Subtraction of Polynomials Monomials and Polynomials ● Addition of Polynomials ● Subtraction of Polynomials ● Evaluating Polynomial Expressions
A LOOK INTO MATH N If you have ever exercised strenuously and then taken your pulse immediately afterward, you may have discovered that your pulse slowed quickly at first and then gradually leveled off. A typical scatterplot of this phenomenon is shown in Figure 5.1(a). These data points cannot be modeled accurately with a line, so a new expression, called a polynomial, is needed to model them. A graph of a polynomial that models these data is shown in Figure 5.1(b) and discussed in Exercise 71. (Source: V. Thomas, Science and Sport.)
NEW VOCABULARY
n Monomial n Degree of a monomial n Coefficient of a monomial n Polynomial n Polynomial in one variable n Binomial n Trinomial n Degree of a polynomial n Like terms
Figure 5.1 Heart Rate After Exercising
10 2 3 4 5 6 7
25
50
75
100
125
150
175
200
t
P
Time (minutes)
(a)
P ul
se (
be at
s pe
r m
in ut
e)
Heart Rate
10 2 3 4 5 6 7
25
50
75
100
125
150
175
200
t
P
Time (minutes)
(b)
P ul
se (
be at
s pe
r m
in ut
e)
Polynomial Model
IS B
N 1-256-49082-2
Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.
3075.2 ADDITION AND SUBTRACTION OF POLYNOMIALS
Monomials and Polynomials A monomial is a number, a variable, or a product of numbers and variables raised to natural number powers. Examples of monomials include
- 3, xy2, 5a2, - z 3, and - 1
2 xy3.
A monomial may contain more than one variable, but monomials do not contain division by variables. For example, the expression 3z is not a monomial. If an expression contains addi- tion or subtraction signs, it is not a monomial.
The degree of a monomial is the sum of the exponents of the variables. If the mono- mial has only one variable, its degree is the exponent of that variable. Remember, when a variable does not have a written exponent, the exponent is implied to be 1. A nonzero number has degree 0, and the number 0 has undefined degree. The number in a monomial is called the coefficient of the monomial. Table 5.2 contains the degree and coefficient of several monomials.
READING CHECK
• How do you determine the degree of a monomial?
A polynomial is a monomial or the sum of two or more monomials. Each monomial is called a term of the polynomial. Addition or subtraction signs separate terms. The expres- sion 2 x 2 - 3x + 5 is a polynomial in one variable with three terms. Examples of polyno- mials in one variable include
- 2 x, 3x + 1, 4y2 - y + 7, and x 5 - 3x 3 + x - 7.
These polynomials have 1, 2, 3, and 4 terms, respectively. A polynomial with two terms is called a binomial, and a polynomial with three terms is called a trinomial.
A polynomial can have more than one variable, as in
x 2y2, 2 xy2 + 5x 2y - 1, and a2 + 2ab + b 2.
Note that all variables in a polynomial are raised to natural number powers. The degree of a polynomial is the degree of the term (or monomial) with greatest degree.
READING CHECK
• How do you determine the degree of a polyomial?
TABLE 5.2 Properties of Monomials
Monomial - 5 6a3b - xy 7y3
Degree 0 4 2 3
Coe fficient - 5 6 - 1 7
EXAMPLE 1 Identifying properties of polynomials
Determine whether the expression is a polynomial. If it is, state how many terms and vari- ables the polynomial contains and give its degree.
(a) 7x 2 - 3x + 1 (b) 5x 3 - 3x 2y3 + xy2 - 2y3 (c) 4 x 2 + 5
x + 1 Solution (a) The expression 7x 2 - 3x + 1 is a polynomial with three terms and one variable. The
first term 7x 2 has degree 2 because the exponent on the variable is 2. The second term - 3x has degree 1 because the exponent on the variable is implied to be 1. The third term 1 has degree 0 because it is a nonzero number. The term with greatest degree is 7x 2, so the polynomial has degree 2.
IS B
N 1-
25 6-
49 08
2- 2
Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.
308 CHAPTER 5 POLYNOMIALS AND EXPONENTS
(b) The expression 5x 3 - 3x 2y3 + xy2 - 2y3 is a polynomial with four terms and two variables, x and y. The first term has degree 3 because the exponent on the variable is 3. The second term has degree 5 because the sum of the exponents on the variables x and y is 5. Likewise, the third term has degree 3 and the fourth term has degree 3. The term with greatest degree is - 3x 2y3, so the polynomial has degree 2 + 3 = 5.
(c) The expression 4 x 2 + 5x + 1 is not a polynomial because it contains division by the polynomial x + 1.
Now Try Exercises 21, 23, 27
Figure 5.2 Adding lw + lw
l lw
w
+ l lw
w
= l 2lw
2w
If two monomials contain the same variables raised to the same powers, we call them like terms. We can add or subtract (combine) like terms but cannot combine unlike terms. The terms lw and 2lw are like terms and can be combined geometrically, as shown in Figure 5.3. If we joined one of the small rectangles with area lw and a larger rect- angle with area 2 lw, then the total area is 3 lw.
The distributive property justifies combining like terms.
1lw + 2lw = (1 + 2)lw = 3lw
The rectangles shown in Figure 5.4 have areas of ab and xy. Together, their area is the sum, ab + xy. However, because these monomials are unlike terms, they cannot be combined into one term.
Figure 5.3 Adding lw + 2lw
l lw 3lw
w
+ l 2lw = l
2w 3w
STUDY TIP
Do you want to know what material will be covered on your next exam? Often, the best place to look is on previ- ously completed assignments and quizzes. If a topic is not discussed in class, is not found on the syllabus, and is not part of your assignments, then your time may be better spent studying other topics.
Figure 5.4 Unlike terms: ab + xy
a ab xy
b
ab
b
+ x
y
xy x
y
= a
Addition of Polynomials Suppose that we have 2 identical rectangles with length l and width w, as illustrated in Figure 5.2. Then the area of one rectangle is lw and the total area is
lw + lw.
This area is equivalent to 2 times lw, which can be expressed as 2lw, or
lw + lw = 2lw.
IS B
N 1-256-49082-2
Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.
3095.2 ADDITION AND SUBTRACTION OF POLYNOMIALS
EXAMPLE 2 Adding like terms
State whether each pair of expressions contains like terms or unlike terms. If they are like terms, add them.
(a) 5x 2, - x 2 (b) 7a2b, 10ab 2 (c) 4rt 2, 1
2 rt 2
Solution (a) The terms 5x 2 and - x 2 have the same variable raised to the same power, so they are like
terms. To add like terms, add their coefficients. Note that the coefficient of - x 2 is - 1.
5x 2 + ( - x 2) = (5 + ( - 1))x 2 Distributive property = 4 x 2 Add.
(b) The terms 7a2b and 10 ab 2 have the same variables, but these variables are not raised to the same powers. They are unlike terms and cannot be added.
(c) The terms 4rt 2 and 12 rt 2 have the same variables raised to the same powers, so they are
like terms. We add them as follows.
4rt 2 + 1 2
rt 2 = a4 + 1 2 brt 2 Distributive property
= 9
2 rt 2 Add.
Now Try Exercises 29, 31, 33
EXAMPLE 3 Adding polynomials
Add by combining like terms. (a) (3x + 4) + ( - 4 x + 2) (b) ( y2 - 2y + 1) + (3y2 + y + 11)
Solution (a) (3x + 4) + (�4 x + 2) = 3x + (�4 x) + 4 + 2 = (3 - 4)x + (4 + 2) = - x + 6
(b) ( y2 - 2y + 1) + (3y2 + y + 11) = y2 + 3y2 - 2y + y + 1 + 11 = (1 + 3)y2 + ( - 2 + 1)y + (1 + 11) = 4y2 - y + 12
Now Try Exercises 37, 39
Recall that the commutative and associative properties of addition allow us to rearrange a sum in any order. For example, if we write each subtraction in 2 x - 5 - 4 x + 10 as addition of the opposite, we have
2 x � 5 � 4 x + 10 = 2 x + (�5) + (�4 x) + 10,
and the terms can be rearranged as
2 x + (�4 x) + (�5) + 10 = 2 x � 4 x � 5 + 10.
If we pay attention to the sign in front of each term in a polynomial, the like terms can be combined without rearranging the terms, as demonstrated in the next example.
To add two polynomials, combine like terms, as illustrated in the next example.
IS B
N 1-
25 6-
49 08
2- 2
Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.
310 CHAPTER 5 POLYNOMIALS AND EXPONENTS
EXAMPLE 4 Adding polynomials
Add (x 3 - 3x 2 + 7x - 4) + (4 x 3 - 5x + 9) by combining like terms.
Solution Remove parentheses and identify like terms with their signs as shown.
x 3 � 3x 2 � 7x � 4 � 4 x 3 � 5x � 9
When like terms (of the same color) are added, the resulting sum is
5x 3 � 3x 2 � 2 x � 5.
Now Try Exercise 41
Polynomials can also be added vertically, as demonstrated in the next example.
EXAMPLE 5 Adding polynomials vertically
Simplify (3x 2 - 3x + 5) + ( - x 2 + x - 6).
Solution Write the polynomials in a vertical format and then add each column of like terms.
3x 2 - 3x + 5 - x 2 + x - 6 2 x 2 - 2 x - 1 Add like terms in each column.
Regardless of the method used, the same answer should be obtained. However, adding ver- tically requires that like terms be placed in the same column.
Now Try Exercise 47
Subtraction of Polynomials To subtract one integer from another, add the first integer and the additive inverse or oppo- site of the second integer. For example, 3 - 5 is evaluated as follows.
3 - 5 = 3 + ( - 5) Add the opposite. = - 2 Simplify.
Similarly, to subtract one polynomial from another, add the first polynomial and the opposite of the second polynomial. To find the opposite of a polynomial, simply negate each term. Table 5.3 lists some polynomials and their opposites.
READING CHECK
• How do you subtract one polynomial from another?
CRITICAL THINKING
What is the result when a polynomial and its opposite are added?
TABLE 5.3 Opposites of Polynomials
Polynomial Opposite
2 x - 4 - 2 x + 4
- x 2 - 2 x + 9 x 2 + 2 x - 9
6 x 3 - 12 - 6 x 3 + 12
- 3x 4 - 2 x 2 - 8 x + 3 3x 4 + 2 x 2 + 8 x - 3
IS B
N 1-256-49082-2
Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.
3115.2 ADDITION AND SUBTRACTION OF POLYNOMIALS
EXAMPLE 6 Subtracting polynomials
Simplify each expression. (a) (3x - 4) - (5x + 1) (b) (5x 2 + 2 x - 3) - (6 x 2 - 7x + 9) (c) (6 x 3 + x 2) - ( - 3x 3 - 9)
Solution (a) To subtract (5x + 1) from (3x - 4), we add the opposite of (5x + 1), or ( - 5x - 1).
(3x - 4) - (5x + 1) = (3x - 4) + (�5x - 1) = (3 � 5)x + (�4 � 1) = �2 x � 5
(b) The opposite of (6 x 2 - 7x + 9) is ( - 6 x 2 + 7x - 9).
(5x 2 + 2 x - 3) - (6 x 2 - 7x + 9) = (5x 2 + 2 x - 3) + ( - 6 x 2 + 7x - 9) = (5 - 6)x 2 + (2 + 7)x + ( - 3 - 9) = - x 2 + 9x - 12
(c) The opposite of ( - 3x 3 - 9) is (3x 3 + 9).
(6 x 3 + x 2) - ( - 3x 3 - 9) = (6 x 3 + x 2) + (3x 3 + 9) = (6 + 3)x 3 + x 2 + 9 = 9x 3 + x 2 + 9
Now Try Exercises 57, 59, 61
NOTE: Some students prefer to subtract one polynomial from another by noting that a sub- traction sign in front of parentheses changes the signs of all of the terms within the paren- theses. For example, part (a) of the previous example could be worked as follows.
(3x - 4) - (5x + 1) = 3x - 4 - 5x - 1 = (3 - 5)x + ( - 4 - 1) = - 2 x - 5
EXAMPLE 7 Subtracting polynomials vertically
Simplify (5x 2 - 2 x + 7) - ( - 3x 2 + 3).
Solution To subtract one polynomial from another vertically, simply add the first polynomial and the opposite of the second polynomial. No x-term occurs in the second polynomial, so insert 0x.
5x 2 - 2 x + 7 3x 2 + 0x - 3 The opposite of - 3x 2 + 3 is 3x 2 - 3 or 3x 2 + 0x - 3. 8 x 2 - 2 x + 4 Add like terms in each column.
Now Try Exercise 69
Evaluating Polynomial Expressions Frequently, monomials and polynomials represent formulas that may be evaluated. We illustrate such applications in the next two examples.
IS B
N 1-
25 6-
49 08
2- 2
Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.
312 CHAPTER 5 POLYNOMIALS AND EXPONENTS
EXAMPLE 8 Writing and evaluating a monomial
Write the monomial that represents the volume of the box having a square bottom, as shown in Figure 5.5. Find the volume of the box if x = 3 feet and y = 2 feet.
Solution The volume of a box is found by multiplying the length, width, and height together. Because the length and width are both x and the height is y, the monomial xxy represents the volume of the box. This can be written x 2y. To calculate the volume, let x = 3 and y = 2 in the monomial x 2y.
x 2y = 32 # 2 = 9 # 2 = 18 cubic feet Now Try Exercise 73
y
x
x
Figure 5.5
EXAMPLE 9 Modeling sales of personal computers
Worldwide sales of personal computers have increased dramatically in recent years, as illustrated in Figure 5.6. The polynomial
0.7868 x 2 + 16.72 x + 122.58
approximates the number of computers sold in millions, where x = 0 corresponds to 2000, x = 1 to 2001, and so on. Estimate the number of personal computers sold in 2008 by using both the graph and the polynomial. (Source: International Data Corporation.)
Solution From the graph shown in Figure 5.7, it appears that personal computer sales were slightly more than 300 million, or about 310 million, in 2008.
Figure 5.6
4 6 8 10
100
0
200
300
400
x
y
Year (0 ↔ 2000)
C om
pu te
r S
al es
( m
il li
on s)
Worldwide Computer Sales
Figure 5.7
4 6 8 10
100
0
200
300
400
x
y
Year (0 ↔ 2000)
C om
pu te
r S
al es
( m
il li
on s)
Worldwide Computer Sales
IS B
N 1-256-49082-2
Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.
3135.2 ADDITION AND SUBTRACTION OF POLYNOMIALS
5.2 Putting It All Together
Monomial A number, variable, or product of numbers and variables raised to natu- ral number powers
The degree is the sum of the exponents.
The coefficient is the number in a monomial.
4 x 2y Degree: 3; coefficient: 4
- 6 x 2 Degree: 2; coefficient: - 6 - a4 Degree: 4; coefficient: - 1
x Degree: 1; coefficient: 1
- 8 Degree: 0; coefficient: - 8
Polynomial A monomial or the sum of two or more monomials
4 x 2 + 8 xy2 + 3y2 Trinomial - 9x 4 + 100 Binomial - 3x 2y3 Monomial
Like Terms Monomials containing the same vari- ables raised to the same powers
10x and - 2 x, 4 x 2 and 3x 2
5ab 2 and - ab 2, 5z and 12 z
CONCEPT EXPLANATION EXAMPLES
Addition of Polynomials To add polynomials, combine like terms.
(x 2 + 3x + 1) + (2 x 2 - 2 x + 7) = (1 + 2)x 2 + (3 - 2)x + (1 + 7) = 3x 2 + x + 8
3xy + 5xy = (3 + 5)xy = 8 xy
Opposite of a Polynomial To obtain the opposite of a polyno- mial, negate each term.
Polynomial Opposite
- 2 x 2 + x - 6 2 x 2 - x + 6 a2 - b 2 - a2 + b 2
- 3x - 18 3x + 18
Subtraction of Polynomials
To subtract one polynomial from another, add the first polynomial and the opposite of the second polynomial.
(x 2 + 3x) - (2 x 2 - 5x) = (x 2 + 3x) + ( - 2 x 2 + 5x) = (1 - 2)x 2 + (3 + 5)x = - x 2 + 8 x
Evaluating a Polynomial To evaluate a polynomial in x, sub- stitute a value for x in the expression and simplify.
To evaluate the polynomial
3x 2 - 2 x + 1 for x = 2,
substitute 2 for x and simplify.
3(2)2 - 2(2) + 1 = 9
The year 2008 corresponds to x = 8 in the given polynomial, so substitute 8 for x and evaluate the resulting expression.
0.7868 x 2 + 16.72 x + 122.58 = 0.7868(8)2 + 16.72(8) + 122.58 � 307 million
The graph and the polynomial give similiar results.
Now Try Exercise 71
IS B
N 1-
25 6-
49 08
2- 2
Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.
314 CHAPTER 5 POLYNOMIALS AND EXPONENTS
5.2 Exercises
CONCEPTS AND VOCABULARY
1. A(n) _____ is a number, a variable, or a product of numbers and variables raised to a natural number power.
2. A(n) _____ is a monomial or a sum of monomials.
3. The _____ of a monomial is the sum of the exponents of the variables.
4. The _____ of a polynomial is the degree of the term with the greatest degree.
5. A polynomial with two terms is called a(n) _____. 6. A polynomial with three terms is called a(n) _____.
7. Two monomials with the same variables raised to the same powers are terms.
8. To add two polynomials, combine terms.
9. To subtract two polynomials, add the first polynomial to the _____ of the second polynomial.
10. Polynomials can be added horizontally or _____.
PROPERTIES OF POLYNOMIALS
Exercises 11–18: Identify the degree and coefficient of the monomial.
11. 3x 2 12. y
13. - ab 14. - 2 xy
15. - 5rt 16. 8 x 2y5
17. 6 18. - 12
Exercises 19–28: Determine whether the expression is a polynomial. If it is, state how many terms and variables the polynomial contains. Then state its degree.
19. - x 20. 7z
21. 4 x 2 - 5x + 9 22. x 3 - 9
23. x + 1 x
24. 5
xy + 1
25. 3x -2y -3 26. 52a3b 4
27. - 23a4bc 28. - 7y -1z-3
Exercises 29–36: State whether the given pair of expres- sions are like terms. If they are like terms, add them.
29. 5x, - 4 x 30. x 2, 8 x 2
31. x 3, - 6 x 3 32. 4 xy, - 9xy
33. 9x, - xy 34. 5x 2y, - 3xy2
35. ab, ba 36. rt 2, - 2t 2r
ADDITION OF POLYNOMIALS
Exercises 37– 46: Add the polynomials.
37. (3x + 5) + ( - 4 x + 4)
38. ( - x + 5) + (2 x - 5)
39. (3x 2 + 4 x + 1) + (x 2 + 4 x)
40. ( - x 2 - x) + (2 x 2 + 3x - 1)
41. ( y3 + 3y2 - 5) + (3y3 + 4y - 4)
42. (4z4 + z2 - 10) + ( - z4 + 4z - 5)
43. ( - xy + 5) + (5xy - 4)
44. (2a2 + b 2) + (3a2 - 5b 2)
45. (a3b 2 + a2b 3) + (a2b 3 - a3b 2)
46. (a2 + ab + b 2) + (a2 - ab + b 2)
Exercises 47–50: Add the polynomials vertically.
47. 4 x 2 - 2 x + 1 5x 2 + 3x - 7
48. 8 x 2 + 3x + 5 - x 2 - 3x - 9
49. - x 2 + x 2 x 2 - 8 x - 1
50. a3 - 3a2b + 3ab 2 - b 3 a3 + 3a2b + 3ab 2 + b 3
SUBTRACTION OF POLYNOMIALS
Exercises 51–56: Write the opposite of the polynomial.
51. 5x 2 52. 17x + 12
53. 3a2 - a + 4
54. - b 3 + 3b
55. - 2t 2 - 3t + 4 56. 7t 2 + t - 10
Exercises 57– 66: Subtract the polynomials.
57. (3x + 1) - ( - x + 3)
58. ( - 2 x + 5) - (x + 7)
59. ( - x 2 + 6 x) - (2 x 2 + x - 2)
IS B
N 1-256-49082-2
Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.
3155.2 ADDITION AND SUBTRACTION OF POLYNOMIALS
60. (2y2 + 3y - 2) - ( y2 - y)
61. (z3 - 2z2 - z) - (4z2 + 5z + 1)
62. (3z4 - z) - ( - z4 + 4z2 - 5)
63. (4 xy + x 2y2) - (xy - x 2y2)
64. (a2 + b 2) - ( - a2 + b 2)
65. (ab 2) - (ab 2 + a3b)
66. (x 2 + 3xy + 4y2) - (x 2 - xy + 4y2)
Exercises 67–70: Subtract the polynomials vertically.
67. (x 2 + 2 x - 3) - (2 x 2 + 7x + 1)
68. (5x 2 - 9x - 1) - (x 2 - x + 3)
69. (3x 3 - 2 x) - (5x 3 + 4 x + 2)
70. (a2 + 3ab + 2b 2) - (a2 - 3ab + 2b 2)
APPLICATIONS
71. Exercise and Heart Rate The polynomial given by 1.6t 2 - 28t + 200 calculates the heart rate shown in Figure 5.1(b) in A Look Into Math for this section, where t represents the elapsed time in minutes since exercise stopped. (a) What is the heart rate when the athlete first stops
exercising? (b) What is the heart rate after 5 minutes? (c) Describe what happens to the heart rate after
exercise stops.
72. Cellular Phone Subscribers In the early years of cel- lular phone technology—from 1986 through 1991— the number of subscribers in millions could be mod- eled by the polynomial 0.163x 2 - 0.146 x + 0.205, where x = 1 corresponds to 1986, x = 2 to 1987, and so on. The graph illustrates this growth.
1987 0
1989 1991
1
2
3
4
5
6
7
x
y
Year
C el
l P
ho ne
S ub
sc ri
be rs
(m il
li on
s)
(a) Use the graph to estimate the number of cellular phone subscribers in 1990.
(b) Use the polynomial to estimate the number of cel- lular phone subscribers in 1990.
(c) Do your answers from parts (a) and (b) agree?
73. Areas of Squares Write a monomial that equals the sum of the areas of the squares. Then calculate this sum for z = 10 inches.
74. Areas of Rectangles Find a monomial that equals the sum of the areas of the three rectangles. Find this sum for a = 5 yards and b = 3 yards.
a
b
a
b
a
b
75. Area of a Figure Find a polynomial that equals the area of the figure. Calculate its area for x = 6 feet.
2x
x
x
x
76. Area of a Rectangle Write a polynomial that gives the area of the rectangle. Calculate its area for x = 3 feet.
7
x
3x
77. Areas of Circles Write a polynomial that gives the sum of the areas of two circles, one with radius x and the other with radius y. Find this sum for x = 2 feet and y = 3 feet. Leave your answer in terms of p.
78. Squares and Circles Write a polynomial that gives
the sum of the areas of a square having sides of length x and a circle having diameter x. Approximate this sum to the nearest hundredth of a square foot for x = 6 feet.
IS B
N 1-
25 6-
49 08
2- 2
Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.
316 CHAPTER 5 POLYNOMIALS AND EXPONENTS
79. World Population The table lists actual and projected world population P in billions for selected years t.
(a) Find the slope of each line segment connecting consecutive data points in the table. Can these data be modeled with a line? Explain.
(b) Does the polynomial 0.077t - 148 give good estimates for the world population in year t? Explain how you decided.
Source: U.S. Census Bureau.
t 1974 1987 1999 2012
P 4 5 6 7
Source: U.S. Postal Service.
t 1963 1975 1987 2002 2007 2011
P 5¢ 13¢ 25¢ 37. 41¢ 44¢
(a) Does the polynomial 0.835t - 1635 model the data in the table exactly?
(b) Does it give approximations that are within 1.5¢ of the actual values?
WRITING ABOUT MATHEMATICS
81. Explain what the terms monomial, binomial, trino- mial, and polynomial mean. Give an example of each.
82. Explain how to determine the degree of a polynomial having one variable. Give an example.
83. Explain how to obtain the opposite of a polynomial. Give an example.
84. Explain how to subtract two polynomials. Give an example.
80. Price of a Stamp The table lists the price P of a first- class postage stamp for selected years t.
Checking Basic ConceptsSECTIONS5.1 and 5.2 1. Evaluate each expression. (a) - 52 (b) 32 - 23
2. Simplify each expression. (a) 103 # 105 (b) (3x 2) ( - 4 x 5) (c) (a3b)2 (d) a x
z3 b4
3. Simplify each expression. (a) (4y3)0 (b) (x 3)2(3x 4)2 (c) 2a2(5a3 - 7)
4. State the number of terms and variables in the polynomial 5x 3y - 2 x 2y + 5. What is its degree?
5. A box has a rectangular bottom twice as long as it is wide.
(a) If the bottom has width w and the box has height h, write a monomial that gives the vol- ume of the box.
(b) Find the volume of the box for w = 12 inches and h = 10 inches.
6. Simplify each expression. (a) (2a2 + 3a - 1) + (a2 - 3a + 7) (b) (4z 3 + 5z) - (2z 3 - 2z + 8) (c) (x 2 + 2 xy + y2) - (x 2 - 2 xy + y2)
5.3 Multiplication of Polynomials Multiplying Monomials ● Review of the Distributive Properties ● Multiplying Monomials and Polynomials ● Multiplying Polynomials
A LOOK INTO MATH N The study of polynomials dates back to Babylonian civilization in about 1800–1600 B.C. Many eighteenth-century mathematicians devoted their entire careers to the study of poly- nomials. Today, polynomials still play an important role in mathematics, often being used to approximate unknown quantities. In this section we discuss the basics of multiplying polynomials. (Source: Historical Topics for the Mathematics Classroom, Thirty-first Yearbook, NCTM.)
IS B
N 1-256-49082-2
Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.
3175.3 MULTIPLICATION OF POLYNOMIALS
Multiplying Monomials A monomial is a number, a variable, or a product of numbers and variables raised to natural number powers. To multiply monomials, we often use the product rule for exponents.
EXAMPLE 1 Multiplying monomials
Multiply. (a) - 5x 2 # 4 x 3 (b) (7xy4)(x 3y2) Solution (a) - 5x 2 # 4 x 3 = ( - 5)(4)x 2x 3 Commutative property = - 20x 2 + 3 The product rule = - 20x 5 Simplify.
(b) (7xy4)(x 3y2) = 7xx 3y4y2 Commutative property = 7x 1 + 3y4 + 2 The product rule = 7x 4y6 Simplify.
Now Try Exercises 9, 13
READING CHECK
• Which rule for exponents is commonly used to multiply monomials?
Review of the Distributive Properties Distributive properties are used frequently for multiplying monomials and polynomials. For all real numbers a, b, and c,
a(b + c) = ab + ac and a(b - c) = ab - ac.
The first distributive property above can be visualized geometrically. For example,
3(x + 2) = 3x + 6
is illustrated in Figure 5.8. The dimensions of the large rectangle are 3 by x + 2, and its area is 3(x + 2). The areas of the two small rectangles, 3x and 6, equal the area of the large rectangle. Therefore 3(x + 2) = 3x + 6.
In the next example we use the distributive properties to multiply expressions.
EXAMPLE 2 Using distributive properties
Multiply. (a) 2(3x + 4) (b) (3x 2 + 4)5 (c) - x(3x - 6)
Solution
(a) 2(3x + 4) = 2 # 3x + 2 # 4 = 6 x + 8 (b) (3x 2 + 4)5 = 3x 2 # 5 + 4 # 5 = 15x 2 + 20 (c) - x(3x - 6) = - x # 3x + x # 6 = - 3x 2 + 6 x Now Try Exercises 15, 19, 21
Figure 5.8 Area: 3x + 6
x + 2
3 3x 6
x 2
Distributive Property 3(x � 2) � 3x � 6
IS B
N 1-
25 6-
49 08
2- 2
Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.
318 CHAPTER 5 POLYNOMIALS AND EXPONENTS
Multiplying Monomials and Polynomials A monomial consists of one term, whereas a polynomial consists of one or more terms separated by + or - signs. To multiply a monomial by a polynomial, we apply the distribu- tive properties and the product rule.
READING CHECK
• What properties are com- monly used to multiply a monomial and a polynomial?
EXAMPLE 3 Multiplying monomials and polynomials
Multiply. (a) 9x(2 x 2 - 3) (b) (5x - 8)x 2 (c) - 7(2 x 2 - 4 x + 6) (d) 4 x 3(x 4 + 9x 2 - 8)
Solution
(a) 9x(2 x 2 - 3) = 9x # 2 x 2 - 9x # 3 Distributive property = 18 x 3 - 27x The product rule
(b) (5x - 8)x 2 = 5x # x 2 - 8 # x 2 Distributive property = 5x 3 - 8 x 2 The product rule
(c) - 7(2 x 2 - 4 x + 6) = - 7 # 2 x 2 + 7 # 4 x - 7 # 6 Distributive property = - 14 x 2 + 28 x - 42 Simplify.
(d) 4 x 3(x 4 + 9x 2 - 8) = 4 x 3 # x 4 + 4 x 3 # 9x 2 - 4 x 3 # 8 Distributive property = 4 x 7 + 36 x 5 - 32 x 3 The product rule
Now Try Exercises 23, 25, 27, 29
We can also multiply monomials and polynomials that contain more than one variable.
EXAMPLE 4 Multiplying monomials and polynomials
Multiply. (a) 2 xy(7x 2y3 - 1) (b) - ab(a2 - b 2)
Solution
(a) 2 xy(7x 2y3 - 1) = 2 xy # 7x 2y3 - 2 xy # 1 Distributive property = 14 xx 2yy3 - 2 xy Commutative property = 14 x 3y4 - 2 xy The product rule
(b) - ab(a2 - b 2) = - ab # a2 + ab # b 2 Distributive property = - aa2b + abb 2 Commutative property = - a3b + ab 3 The product rule
Now Try Exercises 31, 35
Multiplying Polynomials Monomials, binomials, and trinomials are examples of polynomials. Recall that a monomial has one term, a binomial has two terms, and a trinomial has three terms. In the next example we multiply two binomials, using both geometric and symbolic techniques.
IS B
N 1-256-49082-2
Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.
3195.3 MULTIPLICATION OF POLYNOMIALS
EXAMPLE 5 Multiplying binomials
Multiply (x + 4)(x + 2) (a) geometrically and (b) symbolically.
Solution (a) To multiply (x + 4)(x + 2) geometrically, draw a rectangle x + 4 long and x + 2
wide, as shown in Figure 5.9(a). The area of this rectangle equals length times width, or (x + 4)(x + 2). The large rectangle can be divided into four smaller rectangles, which have areas of x 2, 4 x, 2 x, and 8, as shown in Figure 5.9(b). Thus
(x + 4)(x + 2) = x 2 � 4 x � 2 x + 8 = x 2 � 6 x + 8.
(b) To multiply (x + 4)(x + 2) symbolically, apply the distributive property two times.
(x � 4) (x + 2) = (x � 4) (x) + (x � 4) (2) = x # x + 4 # x + x # 2 + 4 # 2 = x 2 � 4 x � 2 x + 8 = x 2 � 6 x + 8
r Now Try Exercise 39
The distributive properties used in part (b) of the previous example show that if we want to multiply (x + 4) by (x + 2), we should multiply every term in x + 4 by every term in x + 2.
(x + 4)(x + 2) = x 2 + 2 x + 4 x + 8 = x 2 + 6 x + 8
NOTE: This process of multiplying binomials is sometimes called FOIL. This acro- nym may be used to remind us to multiply the first terms (F ), outside terms (O), inside terms (I ), and last terms (L). The FOIL process is a shortcut for the process used in Example 5(b).
Multiply the First terms to obtain x 2. (x + 4)(x + 2)
Multiply the Outside terms to obtain 2 x. (x + 4)(x + 2)
Multiply the Inside terms to obtain 4 x. (x + 4)(x + 2)
Multiply the Last terms to obtain 8. (x + 4)(x + 2)
MULTIPLYING POLYNOMIALS
The product of two polynomials may be found by multiplying every term in the first polynomial by every term in the second polynomial and then combining like terms.
READING CHECK
• What kind of polynomials can be multiplied using the FOIL method?
Figure 5.9
x + 2
x + 4
(a) Area = (x + 4)(x + 2)
2 2x
x2
8
x 4
4xx
(b) Area = x2 + 4x + 2x + 8
The following statement summarizes how to multiply two polynomials in general.
IS B
N 1-
25 6-
49 08
2- 2
Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.
320 CHAPTER 5 POLYNOMIALS AND EXPONENTS
EXAMPLE 6 Multiplying binomials
Multiply. Draw arrows to show how each term is found. (a) (3x + 2)(x + 1) (b) (1 - x)(1 + 2 x) (c) (4 x - 3)(x 2 - 2 x)
Solution
(a) (3x + 2)(x + 1) = 3x # x + 3x # 1 + 2 # x + 2 # 1 = 3x 2 � 3x � 2 x + 2 = 3x 2 � 5x + 2
(b) (1 - x)(1 + 2 x) = 1 # 1 + 1 # 2 x - x # 1 - x # 2 x = 1 � 2 x � x - 2 x 2
= 1 � x - 2 x 2
(c) (4 x - 3)(x 2 - 2 x) = 4 x # x 2 - 4 x # 2 x - 3 # x 2 + 3 # 2 x = 4 x 3 � 8 x 2 � 3x 2 + 6 x = 4 x 3 � 11x 2 + 6 x
Now Try Exercises 51, 53, 59
The FOIL process may be helpful for remembering how to multiply two binomials, but it cannot be used for every product of polynomials. In the next example, the general process for multiplying polynomials is used to find products of binomials and trinomials.
EXAMPLE 7 Multiplying polynomials
Multiply. (a) (2 x + 3)(x 2 + x - 1) (b) (a - b)(a2 + ab + b 2) (c) (x 4 + 2 x 2 - 5)(x 2 + 1)
Solution (a) Multiply every term in (2 x + 3) by every term in (x 2 + x - 1).
(2 x + 3)(x 2 + x - 1) = 2 x # x 2 + 2 x # x - 2 x # 1 + 3 # x 2 + 3 # x - 3 # 1 = 2 x 3 � 2 x 2 � 2 x � 3x 2 � 3x - 3 = 2 x 3 � 5x 2 � x - 3
(b) (a - b)(a2 + ab + b 2) = a # a2 + a # ab + a # b 2 - b # a2 - b # ab - b # b 2 = a3 � a2b � ab 2 � a2b � ab 2 - b 3
= a3 - b 3
(c) (x 4 + 2 x 2 - 5)(x 2 + 1) = x 4 # x 2 + x 4 # 1 + 2 x 2 # x 2 + 2 x 2 # 1 - 5 # x 2 - 5 # 1 = x 6 � x 4 � 2 x 4 � 2 x 2 � 5x 2 - 5 = x 6 � 3x 4 � 3x 2 - 5
Now Try Exercises 63, 67, 69
STUDY TIP
Even if you know exactly how to do a math problem correctly, a simple computational error will often cause you to get an incorrect answer. Be sure to take your time on simple calculations.
IS B
N 1-256-49082-2
Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.
3215.3 MULTIPLICATION OF POLYNOMIALS
Polynomials can be multiplied vertically in a manner similar to multiplication of real numbers. For example, multiplication of 123 times 12 is performed as follows.
1 2 3
* 1 2 2 4 6
1 2 3
1 4 7 6
A similar method can be used to multiply polynomials vertically.
EXAMPLE 8 Multiplying polynomials vertically
Multiply 2 x 2 - 4 x + 1 and x + 3 vertically.
Solution Write the polynomials vertically. Then multiply every term in the first polynomial by each term in the second polynomial. Arrange the results so that like terms are in the same column.
2 x 2 - 4 x + 1 x + 3
6 x 2 - 12 x + 3 2 x 3 - 4 x 2 + x 2 x 3 + 2 x 2 - 11x + 3
Multiply top polynomial by 3. Multiply top polynomial by x.
Add each column.
Now Try Exercise 71
MAKING CONNECTIONS
Vertical and Horizontal Formats
Whether you decide to add, subtract, or multiply polynomials vertically or horizontally, remember that the same answer is obtained either way.
EXAMPLE 9 Finding the volume of a box
A box has a width 3 inches less than its height and a length 4 inches more than its height. (a) If h represents the height of the box, write a polynomial that represents the volume of
the box. (b) Use this polynomial to calculate the volume of the box if h = 10 inches.
Solution (a) If h is the height, then h - 3 is the width and h + 4 is the length, as illustrated in
Figure 5.10. Its volume equals the product of these three expressions.
h(h - 3)(h + 4) = (h 2 - 3h)(h + 4)
= h 2 # h + h 2 # 4 - 3h # h - 3h # 4 = h 3 + 4h 2 - 3h 2 - 12h = h 3 + h 2 - 12h
(b) If h = 10, then the volume is
103 + 102 - 12(10) = 1000 + 100 - 120 = 980 cubic inches.
Now Try Exercise 79
Figure 5.10
h
h – 3
h + 4
IS B
N 1-
25 6-
49 08
2- 2
Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.
322 CHAPTER 5 POLYNOMIALS AND EXPONENTS
5.3 Putting It All Together
Distributive Properties For all real numbers a, b, and c,
a(b + c) = ab + ac and a(b - c) = ab - ac.
5(x + 3) = 5x + 15, 3(x - 6) = 3x - 18, and - 2 x(3 - 5x 3) = - 6 x + 10x 4
Multiplying Polynomials The product of two polynomials may be found by multiplying every term in the first polynomial by every term in the second polynomial and then com- bining like terms.
3x(5x 2 + 2 x - 7) = 3x # 5x 2 + 3x # 2 x - 3x # 7 = 15x 3 + 6 x 2 - 21x
(x + 2)(7x - 3) = x # 7x - x # 3 + 2 # 7x - 2 # 3 = 7x 2 - 3x + 14 x - 6 = 7x 2 + 11x - 6
CONCEPT EXPLANATION EXAMPLES
5.3 Exercises
CONCEPTS AND VOCABULARY
1. The equation x 2 # x 3 = x 5 illustrates what rule of exponents?
2. The equation 3(x - 2) = 3x - 6 illustrates what property?
3. The product of two polynomials may be found by multiplying every in the first polynomial by every in the second polynomial and then com- bining like terms.
4. Polynomials can be multiplied horizontally or _____.
MULTIPLICATION OF MONOMIALS
Exercises 5–14: Multiply.
5. x 2 # x 5 6. - a # a5 7. - 3a # 4a 8. 7x # 5x 9. 4 x 3 # 5x 2 10. 6b 6 # 3b 5 11. xy2 # 4 xy 12. 3ab # ab 2 13. ( - 3xy2)(4 x 2y) 14. ( - r 2t 2)( - r 3t)
MULTIPLICATION OF MONOMIALS AND POLYNOMIALS
Exercises 15–36: Multiply and simplify the expression.
15. 3(x + 4) 16. - 7(4 x - 1)
19. (4 - z)z 20. 3z(1 - 5z)
17. - 5(9x + 1)
18. 10(1 - 6 x)
21. - y(5 + 3y)
22. (2y - 8)2y
23. 3x(5x 2 - 4) 24. - 6 x(2 x 3 + 1)
27. - 8(4t 2 + t + 1)
28. 7(3t 2 - 2t - 5)
31. xy(x + y)
32. ab(2a - 3b)
25. (6 x - 6)x 2
26. (1 - 2 x 2)3x 2
29. n2( - 5n2 + n - 2)
30. 6n3(2 - 4n + n2)
33. x 2(x 2y - xy2)
34. 2y2(xy - 5)
35. - ab(a3 - 2b 3)
36. 5rt(r 2 + 2rt + t 2)
IS B
N 1-256-49082-2
Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.
3235.3 MULTIPLICATION OF POLYNOMIALS
MULTIPLICATION OF POLYNOMIALS
Exercises 37–42: (Refer to Example 5.) Multiply the given expression (a) geometrically and (b) symbolically.
37. x(x + 3)
38. 2 x(x + 5)
39. (x + 2)(x + 2)
40. (x + 1)(x + 3)
41. (x + 3)(x + 6)
42. (x + 5)(x + 2)
43. (x + 3)(x + 5)
44. (x - 4)(x - 7)
45. (x - 8)(x - 9)
46. (x + 10)(x + 10)
51. (10y + 7)( y - 1) 52. ( y + 6)(2y + 7)
57. (x - 1)(x 2 + 1)
58. (x + 2)(x 2 - x)
47. (3z - 2)(2z - 5) 48. (z + 6)(2z - 1)
53. (5 - 3a)(1 - 2a) 54. (4 - a)(5 + 3a)
59. (x 2 + 4)(4 x - 3)
60. (3x 2 - 1)(3x 2 + 1)
49. (8b - 1)(8b + 1) 50. (3t + 2)(3t - 2)
55. (1 - 3x)(1 + 3x) 56. (10 - x)(5 - 2 x)
61. (2n + 1)(n2 + 3) 62. (2 - n2)(1 + n2)
Exercises 43–70: Multiply and simplify the expression.
63. (m + 1)(m2 + 3m + 1)
64. (m - 2)(m2 - m + 5)
65. (3x - 2)(2 x 2 - x + 4)
66. (5x + 4)(x 2 - 3x + 2)
67. (x + 1)(x 2 - x + 1)
68. (x - 2)(x 2 + 4 x + 4)
69. (4b 2 + 3b + 7)(b 2 + 3)
70. ( - 3a2 - 2a + 1)(3a2 - 3)
Exercises 71–76: Multiply the polynomials vertically.
71. (x + 2)(x 2 - 3x + 1)
72. (2y - 3)(3y2 - 2y - 2)
73. (a - 2)(a2 + 2a + 4)
74. (b - 3)(b 2 + 3b + 9)
75. (3x 2 - x + 1)(2 x 2 + 1)
76. (2 x 2 - 3x - 5)(2 x 2 + 3)
77. Thinking Generally If a polynomial with m terms and a polynomial with n terms are multiplied, how many terms are there in the product before like terms are combined?
78. Thinking Generally When a polynomial with m terms is multiplied by a second polynomial, the product contains k terms before like terms are combined. How many terms does the second polynomial contain?
APPLICATIONS
79. Volume of a Box (Refer to Example 9.) A box has a width 4 inches less than its height and a length 2 inches more than its height. (a) If h is the height of the box, write a polynomial
that represents the volume of the box. (b) Use this polynomial to calculate the volume for
h = 25 inches.
80. Surface Area of a Box Use the drawing of the box to write a polynomial that represents each of the following.
b + 1
2b + 1
3b
(a) The area of its bottom (b) The area of its front (c) The area of its right side (d) The total area of its six sides
81. Perimeter of a Pen A rectangular pen for a pet has a perimeter of 100 feet. If one side of the pen has length x, then its area is given by x(50 - x). (a) Multiply this expression. (b) Evaluate the expression obtained in part (a) for
x = 25.
82. Rectangular Garden A rectangular garden has a perimeter of 500 feet. (a) If one side of the garden has length x, then write
a polynomial expression that gives its area. Mul- tiply this expression completely.
(b) Evaluate the expression for x = 50 and interpret your answer.
83. Surface Area of a Cube Write a polynomial that rep- resents the total area of the six sides of the cube hav- ing edges with length x + 1.
x + 1
x + 1 x + 1
84. Surface Area of a Sphere The surface area of a sphere with radius r is 4pr 2. Write a polynomial that gives the surface area of a sphere with radius x + 2. Leave your answer in terms of p.
IS B
N 1-
25 6-
49 08
2- 2
Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.
324 CHAPTER 5 POLYNOMIALS AND EXPONENTS
85. Toy Rocket A toy rocket is shot straight up into the air. Its height h in feet above the ground after t sec- onds is represented by the expression t(64 - 16t). (a) Multiply this expression. (b) Evaluate the expression obtained in part (a) and
the given expression for t = 2. (c) Are your answers in part (b) the same? Should
they be the same?
86. Toy Rocket on the Moon (Refer to the preceding exercise.) If the same toy rocket were flown on the moon, then its height h in feet after t seconds would be t164 - 52 t2. (a) Multiply this expression. (b) Evaluate the expression obtained in part (a) and
the given expression for t = 2. Did the rocket go higher on the moon?
WRITING ABOUT MATHEMATICS
87. Explain how the acronym FOIL relates to multiplying two binomials, such as x + 3 and 2 x + 1.
88. Does the FOIL method work for multiplying a bino- mial and a trinomial? Explain.
89. Explain in words how to multiply any two polynomi- als. Give an example.
90. Give two properties of real numbers that are used for multiplying 3x(5x 2 - 3x + 2). Explain your answer.
Working with Real DataGroup Activity
Biology Some types of worms have a remarkable ability to live without moisture. The following table from one study shows the number of worms W surviv- ing after x days without moisture.
(a) Use the equation W = - 0.0014 x 2 - 0.076 x + 50 to find W for each x-value in the table.
Directions: Form a group of 2 to 4 people. Select someone to record the group’s responses for this activity. All members of the group should work cooperatively to answer the questions. If your instructor asks for your results, each member of the group should be prepared to respond.
(b) Discuss how well this equation approximates the data.
(c) Use this equation to estimate the number of worms on day 60 and on day 180. Which answer is most accurate? Explain.
Source: D. Brown and P. Rothery, Models in Biology.
x (days) 0 20 40 80 120 160
W (worms) 50 48 45 36 20 3
5.4 Special Products Product of a Sum and Difference ● Squaring Binomials ● Cubing Binomials
A LOOK INTO MATH N Polynomials are often used to approximate real-world phenomena in applications. Poly- nomials have played an important role in the development of everyday products such as tablet computers, cell phones, and automobiles. Even digital images in computers and interest calculations at a bank make use of polynomials. In this section we discuss how to multiply some special types of binomials.
Product of a Sum and Difference Products of the form (a + b)(a - b) occur frequently in mathematics. Other examples include
(x + y)(x - y) and (2r + 3t)(2r - 3t).
IS B
N 1-256-49082-2
Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.
3255.4 SPECIAL PRODUCTS
These products can always be multiplied by using the techniques discussed in Section 5.3. However, there is a faster way to multiply these special products.
(a + b)(a - b) = a # a � a # b � b # a - b # b = a2 � ab � ba - b 2
= a2 - b 2
In words, the product of a sum of two numbers and their difference equals the difference of their squares. We generalize this method as follows.
STUDY TIP
In mathematics, there are often several correct ways to perform a particular process. If your instructor does not require you to use a specified method, choose the one that works best for you.
READING CHECK
• Explain in words how you can find the product of the sum of two numbers and their difference.
PRODUCT OF A SUM AND DIFFERENCE
For any real numbers a and b,
(a + b)(a - b) = a2 - b 2.
EXAMPLE 1 Finding products of sums and differences
Multiply. (a) (x + y)(x - y) (b) (z - 2)(z + 2) (c) (2r + 3t)(2r - 3t) (d) (5m2 - 4n2)(5m2 + 4n2)
Solution (a) If we let a = x and b = y, then we can apply the rule
(a + b)(a - b) = a2 - b 2.
Thus
(x + y)(x - y) = (x)2 - ( y)2
= x 2 - y2.
(b) Because the expressions (z + 2)(z - 2) and (z - 2)(z + 2) are equal by the commu- tative property, we can apply the formula for the product of a sum and difference.
(z - 2)(z + 2) = (z)2 - (2)2
= z2 - 4
(c) Let a = 2r and b = 3t. Then the product can be evaluated as follows.
(2r + 3t)(2r - 3t) = (2r)2 - (3t)2
= 4r 2 - 9t 2
(d) (5m2 - 4n2)(5m2 + 4n2) = (5m2)2 - (4n2)2 = 25m4 - 16n4
Now Try Exercises 7, 13, 17
The next example demonstrates how the product of a sum and difference can be used to find some products of numbers mentally.
IS B
N 1-
25 6-
49 08
2- 2
Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.
326 CHAPTER 5 POLYNOMIALS AND EXPONENTS
EXAMPLE 2 Finding a product
Use the product of a sum and difference to find 22 # 18. Solution Because 22 = 20 + 2 and 18 = 20 - 2, rewrite and evaluate 22 # 18 as follows. 22 # 18 = (20 + 2)(20 - 2) Rewrite 22 as 20 + 2 and 18 as 20 - 2. = 202 - 22 Product of a sum and difference = 400 - 4 Evaluate exponents. = 396 Subtract.
Now Try Exercise 21
Squaring Binomials Because each side of the square shown in Figure 5.11 has length (a + b), its area equals
(a + b)(a + b),
which can be written as (a + b)2. We can multiply this expression as follows.
(a + b)2 = (a + b)(a + b) = a2 � ab � ba + b 2
= a2 � 2ab + b 2
This result is illustrated geometrically in Figure 5.12, where the area of the large square is (a + b)2. This area can also be found by adding the areas of the four small rectangles.
a2 � ab � ba + b 2 = a2 � 2ab + b 2
The geometric and symbolic results are the same. Note that to obtain the middle term, 2ab, we can multiply the two terms, a and b, in the binomial and double the result.
A similar product that is also the square of a binomial can be calculated as
(a - b)2 = (a - b)(a - b) = a2 � ab � ba + b 2
= a2 � 2ab + b 2.
These results are summarized as follows.
READING CHECK
• Explain in words how you can square a binomial.
SQUARING A BINOMIAL
For any real numbers a and b,
(a + b)2 = a2 + 2ab + b 2 and (a - b)2 = a2 - 2ab + b 2.
That is, the square of a binomial equals the square of the first term, plus (or minus) twice the product of the two terms, plus the square of the last term.
NOTE: (a + b)2 3 a2 + b 2. Do not forget the middle term when squaring a binomial.
EXAMPLE 3 Squaring a binomial
Multiply. (a) (x + 3)2 (b) (2 x - 5)2 (c) (1 - 5y)2 (d) (7a2 + 3b)2
Figure 5.11
a + b
a + b
Area � (a � b)2
Figure 5.12
a
b
a2
ba
ab
a b
b2
(a � b)2 � a2 � 2ab � b 2
IS B
N 1-256-49082-2
Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.
3275.4 SPECIAL PRODUCTS
Solution (a) Let a = x and b = 3 in the formula (a + b)2 = a2 + 2ab + b 2.
(x + 3)2 = (x)2 + 2(x)(3) + (3)2
= x 2 + 6 x + 9
(b) Apply the formula (a - b)2 = a2 - 2ab + b 2 with a = 2 x and b = 5.
(2 x - 5)2 = (2 x)2 - 2(2 x)(5) + (5)2
= 4 x 2 - 20x + 25
(c) (1 - 5y)2 = (1)2 - 2(1)(5y) + (5y)2 = 1 - 10y + 25y2
(d) (7a2 + 3b)2 = (7a2)2 + 2(7a2)(3b) + (3b)2 = 49a4 + 42a2b + 9b 2
Now Try Exercises 27, 29, 35, 39
MAKING CONNECTIONS
Multiplying Binomials and Special Products
If you forget these special products, you can still multiply polynomials by using earlier techniques. For example, the binomial in Example 3(b) can be multiplied as
(2 x - 5)2 = (2 x - 5)(2 x - 5)
= 2 x # 2 x - 2 x # 5 - 5 # 2 x + 5 # 5 = 4 x 2 - 10x - 10x + 25 = 4 x 2 - 20x + 25.
N REAL-WORLD CONNECTION NASA first developed digital pictures because they were easy to transmit through space and because they provided clear images. A digital image from outer space is shown in Figure 5.13.
Today, digital cameras are readily available, and the Internet uses digital images exclu- sively. The next example shows how polynomials relate to digital pictures.
EXAMPLE 4 Calculating the size of a digital picture
A digital picture comprises tiny square units called pixels. Shading individual pixels creates a picture. A simplified version of a digital picture of the letter T is shown in Figure 5.14. This picture includes an image of the letter T that measures 3 pixels by 3 pixels and a 1-pixel border. (a) If a square digital image measures x pixels by x pixels, then a picture that includes the
image and a 1-pixel border will measure x + 2 pixels by x + 2 pixels. Find a polyno- mial that gives the total number of pixels in the picture, including the border.
(b) Let x = 3 and evaluate the polynomial. Does it agree with Figure 5.14?
Solution (a) The total number of pixels equals (x + 2) times (x + 2), or (x + 2)2.
(x + 2)2 = x 2 + 4 x + 4
(b) For x = 3, the polynomial evaluates to 32 + 4 # 3 + 4 = 25, the total number of pixels. This result agrees with Figure 5.14, which has a total of 5 # 5 = 25 pixels with a 3 pixel by 3 pixel image of the letter T inside.
Now Try Exercise 79
Figure 5.13 Digital Picture
Figure 5.14
IS B
N 1-
25 6-
49 08
2- 2
Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.
328 CHAPTER 5 POLYNOMIALS AND EXPONENTS
Cubing Binomials To calculate the volume of the cube shown in Figure 5.15, we find the product of its length, width, and height. Because all sides have the same measure, its volume is (x + 2)3. That is, the volume equals the cube of x + 2.
To multiply the expression (x + 2)3, we proceed as follows.
(x + 2)3 = (x + 2)(x � 2)2 a3 = a # a2
= (x + 2)(x 2 � 4 x � 4) Square the binomial.
= x # x 2 + x # 4 x + x # 4 + 2 # x 2 + 2 # 4 x + 2 # 4 Multiply the polynomials. = x 3 + 4 x 2 + 4 x + 2 x 2 + 8 x + 8 Simplify terms. = x 3 + 6 x 2 + 12 x + 8 Combine like terms.
Figure 5.15 Volume = (x + 2)3
x + 2
x + 2 x + 2
EXAMPLE 5 Cubing a binomial
Multiply (2z - 3)3.
Solution
(2z - 3)3 = (2z - 3)(2z � 3)2 a3 = a # a2
= (2z - 3)(4z2 � 12z � 9) Square the binomial.
= 8z3 - 24z2 + 18z - 12z2 + 36z - 27 Multiply the polynomials. = 8z3 - 36z2 + 54z - 27 Combine like terms.
NOTE: (2z - 3)3 3 (2z)3 - (3)3 = 8z3 - 27.
Now Try Exercise 47
CRITICAL THINKING
Suppose that a student is convinced that the expressions
(x + y)3 and x3 + y 3
are equal. How could you convince the student otherwise?
EXAMPLE 6 Calculating interest
If a savings account pays x percent annual interest, where x is expressed as a decimal, then after 3 years a sum of money will grow by a factor of (1 + x)3. (a) Multiply this expression. (b) Evaluate the expression for x = 0.10 (or 10%), and interpret the result.
Solution (a) (1 + x)3 = (1 + x)(1 + x)2 a3 = a # a2 = (1 + x)(1 + 2 x + x 2) Square the binomial. = 1 + 2 x + x 2 + x + 2 x 2 + x 3 Multiply the polynomials. = 1 + 3x + 3x 2 + x 3 Combine like terms.
IS B
N 1-256-49082-2
Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.
3295.4 SPECIAL PRODUCTS
(b) Let x = 0.1 in the expression 1 + 3x + 3x 2 + x 3.
1 + 3(0.1) + 3(0.1)2 + (0.1)3 = 1.331
The sum of money will increase by a factor of 1.331. For example, $1000 deposited in this account will grow to $1331 in 3 years.
Now Try Exercise 75
5.4 Putting It All Together
Product of a Sum and Difference
For any real numbers x and y,
(x + y)(x - y) = x 2 - y2. (x + 6)(x - 6) = x 2 - 36, (2 x - 3)(2 x + 3) = 4 x 2 - 9, and (x 2 + y2)(x 2 - y2) = x 4 - y4
Squaring a Binomial For all real numbers x and y,
(x + y)2 = x 2 + 2 xy + y2 and (x - y)2 = x 2 - 2 xy + y2.
(x + 4)2 = x 2 + 8 x + 16, (5x - 2)2 = 25x 2 - 20x + 4, and (x 2 + y2)2 = x 4 + 2 x 2y2 + y4
Cubing a Binomial Multiply the binomial by its square. (x + 3)3
= (x + 3)(x + 3)2
= (x + 3)(x 2 + 6 x + 9) = x 3 + 6 x 2 + 9x + 3x 2 + 18 x + 27 = x 3 + 9x 2 + 27x + 27
CONCEPT EXPLANATION EXAMPLES
5.4 Exercises
CONCEPTS AND VOCABULARY
1. (a + b)(a - b) = _____
2. (a + b)2 = _____
3. (a - b)2 = _____
4. (a + b)3 = (a + b) # _____ 5. (True or False?) The two expressions (x + y)2 and
x 2 + y2 are equal for all real numbers x and y.
6. (True or False?) The two expressions (r - t)2 and r 2 - t 2 are equal for all real numbers r and t.
PRODUCT OF A SUM AND DIFFERENCE
Exercises 7–20: Multiply.
7. (x - 3)(x + 3)
8. (x + 6)(x - 6)
9. (4 x - 1)(4 x + 1) 10. (10x + 3)(10x - 3)
11. (1 + 2a)(1 - 2a) 12. (4 - 9b)(4 + 9b)
13. (2 x + 3y)(2 x - 3y) 14. (5r - 6t)(5r + 6t)
15. (ab - 5)(ab + 5)
16. (2 xy + 7)(2 xy - 7)
17. (a2 - b 2)(a2 + b 2)
18. (3x 2 + y2)(3x 2 - y2)
19. (x 3 - y3)(x 3 + y3)
20. (2a4 + b 4)(2a4 - b 4)
Exercises 21–26: (Refer to Example 2.) Use the product of a sum and a difference to evaluate the expression.
21. 101 # 99 22. 52 # 48 23. 23 # 17 24. 29 # 31 25. 90 # 110 26. 38 # 42
IS B
N 1-
25 6-
49 08
2- 2
Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.
330 CHAPTER 5 POLYNOMIALS AND EXPONENTS
SQUARING BINOMIALS
Exercises 27– 40: Multiply.
27. (a - 2)2
28. (x - 7)2
31. (3b + 5)2 32. (7t + 10)2
35. (1 - b)2 36. (1 - 4a)2
29. (2 x + 3)2
30. (7x - 2)2
33. 134 a - 422
37. (5 + y3)2 38. (9 - 5x 2)2
34. 115 a + 122
39. (a2 + b)2
40. (x 3 - y3)2
CUBING BINOMIALS
Exercises 41– 48: Multiply.
41. (a + 1)3
42. (b + 4)3
43. (x - 2)3
44. ( y - 7)3
45. (2 x + 1)3
46. (4z + 3)3
47. (6u - 1)3 48. (5v + 3)3
MULTIPLICATION OF POLYNOMIALS
Exercises 49–66: Multiply, using any appropriate method.
(b) Find the sum of the areas of the smaller rectan- gles inside the large square.
49. 4(5x + 9) 50. (2 x + 1)(3x - 5)
51. (x - 5)(x + 7)
52. (x + 10)(x + 10)
53. (3x - 5)2 54. (x - 3)(x + 9)
55. (5x + 3)(5x + 4) 56. - x 3(x 2 - x + 1)
57. (4b - 5)(4b + 5) 58. (x + 5)3
59. - 5x(4 x 2 - 7x + 2)
60. (4 x 2 - 5)(4 x 2 + 5)
61. (4 - a)3 62. 2 x(x - 3)3
63. x(x + 3)2
64. (x - 1)2(x + 1)
65. (x + 2)(x - 2)(x + 1)(x - 1)
66. (x - y)(x + y)(x 2 + y2)
67. Thinking Generally Multiply (an + b n)(an - b n). 68. Thinking Generally Multiply (an + b n)2.
APPLICATIONS
Exercises 69–72: Do each part and verify that your answers are the same.
(a) Find the area of the large square by multiplying its length and width.
69. 2
x
x 2
70. 4
a
a 4
71.
2x
3
2x 3
72. x
3y
3y x
Exercises 73 and 74: Find a polynomial that represents the following.
(a) The outside surface area given by the six sides of the cube
(b) The volume of the cube
73.
x + 5
x + 5 x + 5
74.
2x + 1
2x + 1 2x + 1
75. Compound Interest (Refer to Example 6.) If a sum of money is deposited in a savings account that is paying x percent annual interest (expressed as a deci- mal), then this sum of money increases by a factor of (1 + x)2 after 2 years. (a) Multiply this expression. (b) Evaluate the polynomial expression found in
part (a) for an annual interest rate of 10%, or x = 0.10, and interpret the answer.
76. Compound Interest If a sum of money is deposited in a savings account that is paying x percent annual interest, then this sum of money increases by a factor of 11 + 1100x23 after 3 years. (a) Multiply this expression.
(b) Evaluate the polynomial expression in part (a) for an annual interest rate of 8%, or x = 8, and interpret the answer.
77. Probability If there is an x percent chance of rain on each of two consecutive days, then the expres- sion (1 - x)2 gives the percent chance that neither day will have rain. Assume that all percentages are expressed as decimals.
IS B
N 1-256-49082-2
Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.
3315.4 SPECIAL PRODUCTS
(a) Multiply this expression. (b) Evaluate the polynomial expression in part (a)
for a 50% chance of rain, or x = 0.50, and inter- pret the answer.
78. Probability If there is an x percent chance of rolling a 6 with one die, then the expression (1 - x)3 gives the percent chance of not rolling a 6 with three dice. Assume that all percentages are expressed as deci- mals or fractions. (a) Multiply this expression. (b) Evaluate the polynomial expression found in part
(a) for a 16.6% chance of rolling a 6, or x = 16 , and interpret the answer.
79. Swimming Pool A square swimming pool has an 8-foot-wide sidewalk around it. (a) If the sides of the pool have length z, as shown in
the accompanying figure, find a polynomial that gives the area of the sidewalk.
(b) Evaluate the polynomial in part (a) for z = 60 and interpret the answer.
8
z
z
80. Digital Picture (Refer to Example 4.) Suppose that a digital picture, including its border, is x + 2 pixels by x + 2 pixels and that the actual image inside the border is x - 2 pixels by x - 2 pixels, as shown in the following figure.
x + 2
x + 2 x – 2
x – 2
(a) Find a polynomial that gives the number of pixels in the border.
(b) Evaluate the polynomial in part (a) for x = 5. (c) Sketch a digital picture of the letter H with
x = 5. Does the picture agree with the answer in part (b)?
(d) Digital pictures typically have large values for x. If a picture has x = 500, find the total number of pixels in its border.
WRITING ABOUT MATHEMATICS
81. Explain why (a + b)2 does not equal a2 + b 2 in general for real numbers a and b.
82. Explain how to find the cube of a binomial.
Checking Basic ConceptsSECTIONS5.3 and 5.4 1. Multiply each expression. (a) ( - 3xy4)(5x 2y) (b) - x(6 - 4 x) (c) 3ab(a2 - 2ab + b 2)
2. Multiply each expression. (a) (x + 3)(4 x - 3) (b) (x 2 - 1)(2 x 2 + 2) (c) (x + y)(x 2 - xy + y2)
3. Multiply each expression. (a) (5x + 2)(5x - 2) (b) (x + 3)2
4. Complete each part and verify that your answers are the same.
(a) Find the area of the large square by squaring the length of one of its sides.
(b) Find the sum of the areas of the smaller rect- angles inside the large square.
5
m
m 5
(c) (2 - 7x)2 (d) (t + 2)3
IS B
N 1-
25 6-
49 08
2- 2
Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.
332 CHAPTER 5 POLYNOMIALS AND EXPONENTS
NEW VOCABULARY
n Scientific notation
5.5 Integer Exponents and the Quotient Rule Negative Integers as Exponents ● The Quotient Rule ● Other Rules for Exponents ● Scientific Notation
A LOOK INTO MATH N In 2009, astronomers discovered a large planet that orbits a distant star. The planet, named WASP-17b, is about 5,880,000,000,000,000 miles from Earth. Also in 2009, the H1N1 virus was identified in a worldwide influenza pandemic. A typical flu virus measures about 0.00000468 inch across. In this section, we will discuss how integer exponents can be used to write such numbers in scientific notation. (Source: Scientific American.)
READING CHECK
• How is a negative integer power on a base related to the corresponding posi- tive integer power on that base?
TABLE 5.4 Powers of 2
Power of 2 Value
21 2
20 1
2-1 12 = 1 21
2-2 14 = 1 22
Decrease exponent by 1
Decrease exponent by 1
Decrease exponent by 1
Divide by 2
Divide by 2
Divide by 2
Table 5.4 shows that 2-1 = 1 21
and 2-2 = 1 22
. In other words, if the exponent on 2 is nega- tive, then the expression is equal to the reciprocal of the corresponding expression with a positive exponent on 2. This discussion suggests the following definition for negative inte- ger exponents.
NEGATIVE INTEGER EXPONENTS
Let a be a nonzero real number and n be a positive integer. Then
a -n = 1
an .
That is, a -n is the reciprocal of an.
STUDY TIP
Mathematics often builds on concepts that have already been studied. Try to get in the regular habit of reviewing topics from earlier parts of the text.
Negative Integers as Exponents So far we have defined exponents that are whole numbers. For example,
50 = 1 and 23 = 2 # 2 # 2 = 8. What if an exponent is a negative integer? To answer this question, consider Table 5.4, which shows values for decreasing powers of 2. Note that each time the exponent on 2 decreases by 1, the resulting value is divided by 2.
IS B
N 1-256-49082-2
Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.
3335.5 INTEGER EXPONENTS AND THE QUOTIENT RULE
EXAMPLE 1 Evaluating negative exponents
Simplify each expression. (a) 2-3 (b) 7-1 (c) x -2 (d) (x + y)-8
Solution
(a) Because a�n = 1
an , 2�3 =
1
23 =
1
2 # 2 # 2 = 1
8 .
(b) 7�1 = 1
71 =
1
7
(c) x�2 = 1
x 2
(d) (x + y)�8 = 1
(x + y)8
Now Try Exercises 7, 19, 25(b)
TECHNOLOGY NOTE
Negative Exponents Calculators can be used to evaluate negative exponents. The figure shows how a graphing cal- culator evaluates the expressions in parts (a) and (b) of Example 1.
2^(�3)�Frac 1/8
7^(�1)�Frac 1/7
CALCULATOR HELP To use the fraction feature (Frac), see Appendix A (pages AP-1 and AP-2).
The rules for exponents discussed in this chapter so far also apply to expressions hav- ing negative exponents. For example, we can apply the product rule, am # an = am + n, as follows.
T Add
2-3 # 22 = 2-3 + 2 = 2-1 = 1 2
We can check this result by evaluating the expression without using the product rule.
2-3 # 22 = 1 23
# 22 = 1 8
# 4 = 4 8 =
1
2
EXAMPLE 2 Using the product rule with negative exponents
Evaluate each expression. (a) 52 # 5-4 (b) 3-2 # 3-1
Solution T
Add
(a) 52 # 5�4 = 52 + (�4) = 5�2 = 1 52
= 1
25
(b) 3-2 # 3-1 = 3-2 + (-1) = 3-3 = 1 33
= 1
27 Now Try Exercise 9
IS B
N 1-
25 6-
49 08
2- 2
Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.
334 CHAPTER 5 POLYNOMIALS AND EXPONENTS
EXAMPLE 3 Using the rules of exponents
Simplify the expression. Write the answer using positive exponents. (a) x 2 # x -5 (b) ( y3)-4 (c) (rt)-5 (d) (ab)-3(a -2b)3
Solution (a) Using the product rule, am # an = am�n, gives
x 2 # x�5 = x 2 + (�5) = x�3 = 1 x 3
.
(b) Using the power rule, (am)n = amn, gives
( y3)�4 = y3(�4) = y�12 = 1
y12 .
(c) Using the power rule, (ab)n = anb n, gives
(rt)�5 = r�5t�5 = 1
r 5 # 1
t 5 =
1
r 5t 5 .
This expression could also be simplified as follows.
(rt)-5 = 1
(rt)5 =
1
r 5t 5
(d) (ab)-3(a -2b)3 = a -3b -3a -6b 3
= a -3 + (-6)b -3 + 3
= a -9b 0
= 1
a9 # 1
= 1
a9
Now Try Exercises 21, 27, 29(a)
The Quotient Rule Consider the division problem
34
32 =
3 # 3 # 3 # 3 3 # 3 =
3 3
# 3 3
# 3 # 3 = 1 # 1 # 32 = 32. Because there are two more 3s in the numerator than in the denominator, the result is
T Subtract
34 - 2 = 32.
That is, to divide exponential expressions having the same base, subtract the exponent of the denominator from the exponent of the numerator and keep the same base. This rule is called the quotient rule, which we express in symbols as follows.
THE QUOTIENT RULE
For any nonzero number a and integers m and n,
am
an = am - n.
IS B
N 1-256-49082-2
Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.
3355.5 INTEGER EXPONENTS AND THE QUOTIENT RULE
EXAMPLE 4 Using the quotient rule
Simplify each expression. Write the answer using positive exponents.
(a) 43
45 (b)
6a7
3a4 (c)
xy7
x 2y5
Solution
T Subtract
(a) 43
45 = 43 - 5 = 4-2 =
1
42 =
1
16
(b) 6a7
3a4 =
6
3 # a7
a4 = 2a7 - 4 = 2a3
(c) xy7
x 2y5 =
x 1
x 2 # y7
y5 = x 1 - 2y7 - 5 = x -1y2 =
y2
x
Now Try Exercises 13(b), 31(b), 33(a)
MAKING CONNECTIONS
The Quotient Rule and Simplifying Quotients
Some quotients can be simplified mentally. Because
x 5
x 3 =
x # x # x # x # x x # x # x ,
the quotient x 5
x 3 has five factors of x in the numerator and three factors of x in the denominator.
There are two more factors of x in the numerator than in the denominator, 5 - 3 = 2, so this expression simplifies to x 2. Similarly,
x 3
x 5 =
x # x # x x # x # x # x # x
has two more factors of x in the denominator than in the numerator. This quotient x 3
x 5 sim-
plifies to 1 x 2
. Use this technique to simplify the expressions
z7
z4 ,
a5
a8 , and
x 6y2
x 3y7 .
Other Rules for Exponents Other rules can be used to simplify expressions with negative exponents.
QUOTIENTS AND NEGATIVE EXPONENTS
The following three rules hold for any nonzero real numbers a and b and positive integers m and n.
1. 1
a -n = an 2.
a -n
b -m =
b m
an 3. aa
b b-n = ab
a bn
IS B
N 1-
25 6-
49 08
2- 2
Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.
336 CHAPTER 5 POLYNOMIALS AND EXPONENTS
We demonstrate the validity of these rules as follows.
1. 1
a -n =
1
1
an
= 1 # a n
1 = an
2. a -n
b -m =
1
an
1
b m
= 1
an # b m
1 =
b m
an
3. aa b b-n = a -n
b -n =
1
an
1
b n
= 1
an # b n
1 =
b n
an = ab
a bn
EXAMPLE 5 Working with quotients and negative exponents
Simplify each expression. Write the answer using positive exponents.
(a) 1
2-5 (b)
3-3
4-2 (c)
5x -4y2
10x 2y -4 (d) a 2
z2 b-4
Solution
(a) 1
2�5 = 25 = 2 # 2 # 2 # 2 # 2 = 32 (b) 3
�3
4�2 =
42
33 =
16
27
(c) 5x�4y2
10x 2y�4 =
y2y4
2 x 2x 4 =
y6
2 x 6 (d) a 2
z2 b�4 = az2
2 b4 = z8
24 =
z8
16
Now Try Exercises 15(b), 17, 37, 47
The rules for natural number exponents that are summarized in Section 5.1 on page 303 also hold for integer exponents. Additional rules for integer exponents are summarized as follows.
RULES FOR INTEGER EXPONENTS
The following rules hold for nonzero real numbers a and b, and positive integers m and n.
Description Rule Example
Negative Exponents (1) a -n = 1
an 9-2 =
1
92 =
1
81
The Quotient Rule am
an = am - n
23
2-2 = 23 - (-2) = 25
Negative Exponents (2) 1
a -n = an
1
7-5 = 75
Negative Exponents (3) a -n
b -m =
b m
an
4-3
2-5 =
25
43
Negative Exponents (4) aa b b - n = ab
a bn a1
5 b - 2 = a5
1 b2 = 25
IS B
N 1-256-49082-2
Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.
3375.5 INTEGER EXPONENTS AND THE QUOTIENT RULE
Scientific Notation Powers of 10 are important because they are used in science to express numbers that are either very small or very large in absolute value. Table 5.5 lists some powers of 10. Note that if the power of 10 decreases by 1, the result decreases by a factor of 110, or equivalently, the decimal point is moved one place to the left. Table 5.6 shows the names of some important powers of 10.
READING CHECK
• What kinds of numbers are often expressed in scientific notation?
TABLE 5.5 Powers of 10
Power of 10 Value
103 1000
102 100
101 10
100 1
10-1 110 = 0.1
10-2 1100 = 0.01
10-3 11000 = 0.001
TABLE 5.6 Important Powers of 10
Power of 10 Name
1012 Trillion
109 Billion
106 Million
103 Thousand
10-1 Tenth
10-2 Hundredth
10-3 Thousandth
10-6 Millionth
Recall that numbers written in decimal notation are sometimes said to be in standard form. Decimal numbers that are either very large or very small in absolute value can be expressed in scientific notation.
TECHNOLOGY NOTE
Powers of 10 Calculators make use of sci- entific notation, as illustrated in the accompanying figure. The letter E denotes a power of 10. That is,
2.5E13 = 2.5 : 1013 and 5E�6 = 5 : 10�6.
25000000000000 2.5E13
.000005 5E�6
Note: The calculator has been set in scientific mode.
REAL-WORLD CONNECTION As mentioned in A Look Into Math for this section, the distance to the planet WASP-17b is about 5,880,000,000,000,000 miles. This distance can be written in scientific notation as 5.88 * 1015 because
5,880,000,000,000,000 = 5.88 * 1015. 8 15 decimal places
The 1015 indicates that the decimal point in 5.88 should be moved 15 places to the right. A typical virus is about 0.00000468 inch in diameter, which can be written in scientific
notation as 4.68 * 10-6 because
0.00000468 = 4.68 * 10-6. (1)1* 6 decimal places
The 10�6 indicates that the decimal point in 4.68 should be moved 6 places to the left. The following definition provides a more complete explanation of scientific notation.
SCIENTIFIC NOTATION
A real number a is in scientific notation when a is written in the form b * 10n, where 1 … 0b 0 6 10 and n is an integer.
IS B
N 1-
25 6-
49 08
2- 2
Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.
338 CHAPTER 5 POLYNOMIALS AND EXPONENTS
EXAMPLE 6 Converting scientific notation to standard form
Write each number in standard form. (a) 5.23 * 104 (b) 8.1 * 10-3 (c) 6 * 10-2
Solution (a) The positive exponent 4 indicates that the decimal point in 5.23 is to be moved 4 places
to the right.
5.23 * 104 = 5. 2 3 0 0. = 52,300 1 2 3 4
(b) The negative exponent - 3 indicates that the decimal point in 8.1 is to be moved 3 places to the left.
8.1 * 10�3 = 0. 0 0 8. 1 = 0.0081 1 2 3
(c) 6 * 10�2 = 0. 0 6. = 0.06 1 2
Now Try Exercises 61, 63
The following steps can be used for writing a positive number a in scientific notation.
NOTE: The scientific notation for a negative number a is the opposite of the scientific nota- tion of 0a 0 . For example, 450 = 4.5 * 102 and - 450 = - 4.5 * 102.
WRITING A POSITIVE NUMBER IN SCIENTIFIC NOTATION
For a positive, rational number a expressed as a decimal, if 1 … a 6 10, then a = a * 100. Otherwise, use the following process to write a in scientific notation.
1. Move the decimal point in a until it becomes a number b such that 1 … b 6 10. 2. Let the positive integer n be the number of places the decimal point was moved. 3. Write a in scientific notation as follows. • If a Ú 10, then a = b * 10n. • If a 6 1, then a = b * 10-n.
EXAMPLE 7 Writing a number in scientific notation
Write each number in scientific notation. (a) 308,000,000 (U.S. population in 2010) (b) 0.001 (Approximate time in seconds for sound to travel one foot)
Solution (a) Move the assumed decimal point in 308,000,000 eight places to obtain 3.08.
3. 0 8 0 0 0 0 0 0.
1 2 3 4 5 6 7 8
Since 308,000,000 Ú 10, the scientific notation is 3.08 * 108.
IS B
N 1-256-49082-2
Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.
3395.5 INTEGER EXPONENTS AND THE QUOTIENT RULE
Numbers in scientific notation can be multiplied by applying properties of real numbers and properties of exponents.
(6 * 104) # (3 * 103) = (6 # 3) * (104 # 103) Properties of real numbers = 18 * 107 Product rule = 1.8 * 108 Scientific notation
Division can also be performed with scientific notation.
6 * 104
3 * 103 =
6 3
* 104
103 Property of fractions
= 2 * 101 Quotient rule
These results are supported in Figure 5.16, where the calculator is in scientific mode.
In the next example we show how to use scientific notation in an application.
CALCULATOR HELP To display numbers in scientific notation, see Appendix A (page AP-2).
Figure 5.16
(6�10^4)(3�10^3)
1.8E8 (6�10^4)/(3�10^3 )
2E1
EXAMPLE 8 Analyzing the cost of Internet advertising
In 2009, a total of $2.38 * 1010 was spent on Internet advertising in the United States. At that time the population of the United States was 3.05 * 108. Determine how much was spent per person on Internet advertising. (Source: New York Times.)
Solution To determine the amount spent per person, divide $2.38 * 1010 by 3.05 * 108.
2.38 * 1010
3.05 * 108 =
2.38
3.05 * 1010 - 8 � 0.78 * 102 = 78
In 2009, about $78 per person was spent on Internet advertising.
Now Try Exercise 97
CRITICAL THINKING
Estimate the number of seconds that you have been alive. Write your answer in scientific notation.
(b) Move the decimal point in 0.001 three places to obtain 1.
0. 0 0 1.
1 2 3
Since 0.001 6 1, the scientific notation is 1 * 10�3.
Now Try Exercises 75, 79
5.5 Putting It All Together
For the rules for integer exponents in this table, assume that a and b are nonzero real num- bers and that m and n are integers.
CONCEPT EXPLANATION EXAMPLES
Negative Integer Exponents a
-n = 1
an 2-4 =
1
24 =
1
16 , a -8 =
1
a8 , and
(xy)-2 = 1
(xy)2 =
1
x 2y2
continued on next page
IS B
N 1-
25 6-
49 08
2- 2
Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.
CHAPTER 5 POLYNOMIALS AND EXPONENTS340
CONCEPT EXPLANATION EXAMPLES
Quotient Rule am
an = am - n 7
2
74 = 72 - 4 = 7-2 =
1
72 =
1
49 and
x 6
x 3 = x 6 - 3 = x 3
Quotients and Negative Integer Exponents
1. 1
a -n = an
2. a -n
b -m =
b m
an
3. aa b b-n = ab
a bn
1. 1
5-2 = 52 = 25
2. x -4
y -2 =
y2
x 4
3. a2 3 b-3 = a3
2 b3 = 33
23 =
27
8
Scientific Notation Write a as b * 10n, where 1 … 0b 0 6 10 and n is an integer. 23,500 = 2.35 * 104,0.0056 = 5.6 * 10-3, and
1000 = 1 * 103
5.5 Exercises
CONCEPTS AND VOCABULARY
Exercises 1–5: Complete the given rule for integer expo- nents m and n, where a and b are nonzero real numbers.
1. a -n = 2. 1
a -n =
3. am
an = 4.
a -n
b -m =
5. aa b b-n =
6. To write a positive number a in scientific notation as b * 10n, the number b must satisfy _____.
NEGATIVE EXPONENTS
Exercises 7–18: Simplify the expression.
7. (a) 4-1 (b) a1 3 b-2
8. (a) 6-2 (b) 2.5-1
9. (a) 23 # 2-2 (b) 10-1 # 10-2
10. (a) 3-4 # 32 (b) 104 # 10-2 11. (a) 3-2 # 3-1 # 3-1 (b) (23)-1 12. (a) 2-3 # 25 # 2-4 (b) (3-2)-2
13. (a) (3243)-1 (b) 45
42
14. (a) (2-232)-2 (b) 55
53
15. (a) 19
17 (b)
1
4-3
16. (a) - 64
6 (b)
1
6-2
17. (a) 5-2
5-4 (b) a2
7 b-2
18. (a) 7-3
7-1 (b) a3
4 b-3
continued from previous page
IS B
N 1-256-49082-2
Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.
3415.5 INTEGER EXPONENTS AND THE QUOTIENT RULE
Exercises 19–50: Simplify the expression. Write the answer using positive exponents.
19. (a) x -1 (b) a -4
20. (a) y -2 (b) z-7
21. (a) x -2 # x -1 # x (b) a -5 # a -2 # a -1 22. (a) y -3 # y4 # y -5 (b) b 5 # b -3 # b -6 23. (a) x 2y -3x -5y6 (b) (xy)-3
24. (a) a -2b -6b 3a -1 (b) (ab)-1
25. (a) (2t)-4 (b) (x + 1)-7
26. (a) (8c)-2 (b) (a + b)-9
27. (a) (a -2)-4 (b) (rt 3)-2
28. (a) (4 x 3)-3 (b) (xy -3)-2
29. (a) (ab)2(a2)-3 (b) x 4
x 2
30. (a) (x 3)-2(xy)4y -5 (b) y9
y5
31. (a) a10
a -3 (b)
4z
2z4
32. (a) b 5
b -2 (b)
12 x 2
24 x 7
33. (a) - 4 xy5
6 x 3y2 (b)
x -4
x -1
34. (a) 12a6b 2
8ab 3 (b)
y -2
y -7
35. (a) 10b -4
5b -5 (b) aa
b b3
36. (a) 8a -2
2a -3 (b) a2 x
y b5
37. (a) 6 x 2y -4
18 x -5y4 (b)
16a -3b -5
4a -8b
38. (a) m2n4
3m-5n4 (b)
7x -3y -5
x -3y -2
39. (a) 1
y -5 (b)
4
2t -3
40. (a) 1
z-6 (b)
5
10b -5
41. (a) 3a4
(2a -2)3 (b)
(2b 5)-3
4b -6
42. (a) (2 x 4)-2
5x -2 (b)
2y5
(3y -4)-2
43. (a) 1
(xy)-2 (b)
1
(a2b)-3
44. (a) 1
(ab)-1 (b)
1
(rt 4)-2
45. (a) (3m4n)-2
(2mn-2)3 (b)
( - 4 x 4y)2
(xy -5)-3
46. (a) (x 4y2)2
( - 2 x 2y -2)3 (b)
(m2n-6)-2
(4m2n-4)-3
47. (a) aa b b-2 (b) a u
4v b-1
48. (a) a2 x y b-3 (b) a5u
3v b-2
49. (a) a 3a4b 2ab -2
b-2 (b) a 4m4n 5m-3n2
b2 50. (a) a 2 x 4y2
3x 3y -3 b3 (b) a a -5b 2
2ab -2 b-2
51. Thinking Generally For positive integers m and n
show that an
am =
1
am - n .
52. Thinking Generally For positive integers m and n
show that a -n
a -m = am - n.
SCIENTIFIC NOTATION
Exercises 53–58: (Refer to Table 5.6.) Write the value of the power of 10 in words.
53. 103 54. 106
55. 109 56. 10-1
57. 10-2 58. 10-6
Exercises 59–70: Write the expression in standard form.
59. 2 * 103 60. 5 * 102
61. 4.5 * 104 62. 7.1 * 106
63. 8 * 10-3 64. 9 * 10-1
65. 4.56 * 10-4 66. 9.4 * 10-2
67. 3.9 * 107 68. 5.27 * 106
69. - 5 * 105 70. - 9.5 * 103
IS B
N 1-
25 6-
49 08
2- 2
Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.
342 CHAPTER 5 POLYNOMIALS AND EXPONENTS
Exercises 71–82: Write the number in scientific notation.
71. 2000 72. 11,000
73. 567 74. 9300
75. 12,000,000 76. 600,000
77. 0.004 78. 0.0008
79. 0.000895 80. 0.0123
81. - 0.05 82. - 0.934
Exercises 83–90: Evaluate the expression. Write the answer in standard form.
83. (5 * 103)(3 * 102)
84. (2.1 * 102)(2 * 104)
85. ( - 3 * 10-3)(5 * 102)
86. (4 * 102)(1 * 103)(5 * 10-4)
87. 4 * 105
2 * 102 88.
9 * 102
3 * 106
89. 8 * 10-6
4 * 10-3 90.
6.3 * 102
2 * 10-3
APPLICATIONS
91. Light-year The distance that light travels in 1 year is called a light-year. Light travels at 1.86 * 105 miles per second, and there are about 3.15 * 107 seconds in 1 year. (a) Estimate the number of miles in 1 light-year. (b) Except for the sun, Alpha Centauri is the near-
est star, and its distance is 4.27 light-years from Earth. Estimate its distance in miles. Write your answer in scientific notation.
92. Milky Way It takes 2 * 108 years for the sun to make one orbit around the Milky Way galaxy. Write this number in standard form.
93. Speed of the Sun (Refer to the two previous exer- cises.) Assume that the sun’s orbit in the Milky Way galaxy is circular with a diameter of 105 light-years. Estimate how many miles the sun travels in 1 year.
94. Distance to the Moon The moon is about 240,000 miles from Earth.
(a) Write this number in scientific notation. (b) If a rocket traveled at 4 * 104 miles per hour,
how long would it take for it to reach the moon? 95. Online Exploration In 1997, the creators of the
Internet search engine BackRub renamed it Google. This new name is a play on the word googol, which is a very large number. Look up a googol and write it in scientific notation.
96. Online Exploration An astronomical unit (AU) is based on the distance from Earth to the sun. Look up the distance in kilometers from Earth to the sun.
(a) Write an astronomical unit in standard form to the nearest million kilometers.
(b) Convert your rounded answer from part (a) to scientific notation.
97. Gross Domestic Product The gross domestic prod- uct (GDP) is the total national output of goods and services valued at market prices within the United States. The GDP of the United States in 2005 was $12,460,000,000,000. (Source: Bureau of Economic Analysis.)
(a) Write this number in scientific notation. (b) In 2005, the U.S. population was 2.98 * 108. On
average, how many dollars of goods and services were produced by each individual?
98. Average Family Net Worth A family refers to a group of two or more people related by birth, mar- riage, or adoption who reside together. In 2000, the average family net worth was $280,000, and there were about 7.2 * 107 families. Calculate the total family net worth in the United States in 2000. (Source: U.S. Census Bureau.)
WRITING ABOUT MATHEMATICS
99. Explain what a negative exponent is and how it is different from a positive exponent. Give an example.
100. Explain why scientific notation is helpful for writ- ing some numbers.
IS B
N 1-256-49082-2
Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.
3435.6 DIVISION OF POLYNOMIALS
Working with Real DataGroup Activity
Water in a Lake East Battle Lake in Minnesota covers an area of about 1950 acres, or 8.5 * 107 square feet, and its average depth is about 3.2 * 101 feet. (a) Estimate the cubic feet of water in the lake. (Hint:
volume = area * average depth.) (b) One cubic foot of water equals about 7.5 gallons.
How many gallons of water are in this lake?
(c) The population of the United States is about 3.1 * 108, and the average American uses about 1.5 * 102 gallons of water per day. Could this lake supply the American population with water for 1 day?
Directions: Form a group of 2 to 4 people. Select a person to record the group’s responses for this activity. All members of the group should work cooperatively to answer the questions. If your instructor asks for your results, each member of the group should be prepared to respond.
5.6 Division of Polynomials Division by a Monomial ● Division by a Polynomial
A LOOK INTO MATH N The study of polynomials has occupied the minds of mathematicians for centuries. During the sixteenth century, Girolamo Cardano and other Italian mathematicians discovered how to solve higher degree polynomial equations. In this section we demonstrate how to divide polynomials. Division is often needed to factor polynomials and to solve polynomial equa- tions. (Source: H. Eves, An Introduction to the History of Mathematics.)
Division by a Monomial To add two fractions with like denominators, we use the property
a
d +
b
d =
a + b d
.
For example, 17 + 3 7 =
1 + 3 7 =
4 7.
To divide a polynomial by a monomial, we use the same property, only in reverse. That is,
a + b d
= a
d +
b
d .
Note that each term in the numerator is divided by the monomial in the denominator. The next example shows how to divide a polynomial by a monomial.
Girolamo Cardano (1501–1576)
EXAMPLE 1 Dividing a polynomial by a monomial
Divide.
(a) a5 + a3
a2 (b)
5x 4 + 10x 10x
(c) 3y2 + 2y - 12
6y
IS B
N 1-
25 6-
49 08
2- 2
Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.
344 CHAPTER 5 POLYNOMIALS AND EXPONENTS
Solution
(a) a5 � a3
a2 =
a5
a2 �
a3
a2 = a5 - 2 + a3 - 2 = a3 + a
(b) 5x 4 � 10x
10x =
5x 4
10x �
10x
10x =
x 3
2 + 1
(c) 3y2 � 2y � 12
6y =
3y2
6y �
2y
6y �
12
6y =
y
2 +
1
3 -
2 y
Now Try Exercises 17, 19, 21
MAKING CONNECTIONS
Division and Simplification
A common mistake made when dividing expressions is to “cancel” incorrectly. Note in Example 1(b) that
5x 4 + 10x 10x
3 5x 4 + 10x
10x .
The monomial must be divided into every term in the numerator.
When dividing two natural numbers, we can check our work by multiplying. For exam- ple, 105 = 2, and we can check this result by finding the product 5 # 2 = 10. Similarly, to check
a5 � a3
a2 = a3 � a
in Example 1(a) we can multiply a2 and a3 � a.
a2(a3 � a) = a2 # a3 + a2 # a Distributive property = a5 � a3 ✓ It checks.
EXAMPLE 2 Dividing and checking
Divide the expression 8 x 3 - 4 x 2 + 6 x
2 x 2 and then check the result.
Solution Be sure to divide 2 x 2 into every term in the numerator.
8 x 3 � 4 x 2 � 6 x
2 x 2 =
8 x 3
2 x 2 �
4 x 2
2 x 2 �
6 x
2 x 2 = 4 x - 2 +
3 x
Check:
2 x 2a4 x - 2 + 3 x b = 2 x 2 # 4 x - 2 x 2 # 2 + 2 x 2 # 3
x
= 8 x 3 - 4 x 2 + 6 x ✓
Now Try Exercise 23
IS B
N 1-256-49082-2
Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.
3455.6 DIVISION OF POLYNOMIALS
EXAMPLE 3 Finding the length of a rectangle
The rectangle in Figure 5.17 has an area A = x 2 + 2 x and width x. Write an expression for its length l in terms of x.
STUDY TIP
Do you have enough time to study your notes and complete your assign- ments? One way to man- age your time is to make a list of your time com- mitments and determine the amount of time that each activity requires. Remember to include time for eating, sleeping, and relaxing!
Figure 5.17
x A = x2 + 2x
l
Solution The area A of a rectangle equals length l times width w, or A = lw. Solving for l gives
l = A w
.
Thus to find the length of the given rectangle, divide the area by the width.
l = x 2 + 2 x
x =
x 2
x +
2 x x
= x + 2
The length of the rectangle is x + 2. The answer checks because x(x + 2) = x 2 + 2 x.
Now Try Exercise 49
To check this result, we find the product of the quotient and divisor and then add the remain- der. Because 67 # 4 + 3 = 271, the answer checks. The quotient and remainder can also be expressed as 6734. Division of polynomials is similar to long division of natural numbers.
Division by a Polynomial To understand division by a polynomial better, we first need to review some terminology related to long division of natural numbers. To compute 271 , 4, we complete long divi- sion as follows.
67 R 3
4�271 24
31
28
3
Quotient h vRemainder Divisorh v Dividend
EXAMPLE 4 Dividing polynomials
Divide 6 x 2 + 13x + 3
3x + 2 and check.
Solution Begin by dividing the first term of 3x + 2 into the first term of 6 x 2 + 13x + 3. That is, divide 3x into 6 x 2 to obtain 2 x. Then find the product of 2 x and 3x + 2, or 6 x 2 + 4 x, place it below 6 x 2 + 13x, and subtract. Bring down the 3.
2x 3x + 2�6 x 2 + 13x + 3
6 x 2 + 4 x 9x + 3
6 x 2
3x = 2 x
2 x(3x + 2) = 6 x 2 + 4 x Subtract: 13x - 4 x = 9x. Bring down the 3.
IS B
N 1-
25 6-
49 08
2- 2
Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.
346 CHAPTER 5 POLYNOMIALS AND EXPONENTS
In the next step, divide 3x into the first term of 9x + 3 to obtain 3. Then find the product of 3 and 3x + 2, or 9x + 6, place it below 9x + 3, and subtract.
2 x + 3 3x + 2�6 x 2 + 13x + 3
6 x 2 + 4 x 9x + 3 9x + 6
- 3
9x 3x
= 3
3(3x + 2) = 9x + 6 Subtract: 3 - 6 = - 3.
The quotient is 2 x + 3 with remainder - 3. This result can also be written as
2 x + 3 + - 3
3x + 2 , Quotient +
Remainder
Divisor
in the same manner that 67 R 3 was written as 6734. Check polynomial division by adding the remainder to the product of the divisor and
the quotient. That is,
(Divisor)(Quotient) + Remainder = Dividend.
For this example, the equation becomes
(3x � 2)(2 x � 3) + (�3) = 3x # 2 x + 3x # 3 + 2 # 2 x + 2 # 3 � 3 = 6 x 2 + 9x + 4 x + 6 - 3 = 6 x 2 � 13x � 3. ✓ It checks.
Now Try Exercise 27
READING CHECK
• How can you check a polynomial division problem?
EXAMPLE 5 Dividing polynomials having a missing term
Simplify (3x 3 + 2 x - 4) , (x - 2).
Solution Because the dividend does not have an x 2-term, insert 0x 2 as a “place holder.” Then begin by dividing x into 3x 3 to obtain 3x 2.
3x 2
x - 2�3x 3 + 0 x 2 + 2 x - 4 3x 3 - 6 x 2
6 x 2 + 2 x
3x 3
x = 3x 2
3x 2(x - 2) = 3x 3 - 6 x 2
Subtract 0x 2 - ( - 6 x 2) = 6 x 2. Bring down 2x.
In the next step, divide x into 6 x 2.
3x 2 + 6 x x - 2�3x 3 + 0 x 2 + 2 x - 4
3x 3 - 6 x 2
6 x 2 + 2 x 6 x 2 - 12 x
14 x - 4
6 x 2
x = 6 x
6 x(x - 2) = 6 x 2 - 12 x Subtract: 2 x - ( - 12 x) = 14 x. Bring down - 4.
IS B
N 1-256-49082-2
Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.
3475.6 DIVISION OF POLYNOMIALS
Now divide x into 14 x.
3x 2 + 6 x + 14 x - 2�3x 3 + 0 x 2 + 2 x - 4
3x 3 - 6 x 2
6 x 2 + 2 x 6 x 2 - 12 x
14 x - 4 14 x - 28
24
The quotient is 3x 2 + 6 x + 14 with remainder 24. This result can also be written as
3x 2 + 6 x + 14 + 24
x - 2 .
Now Try Exercise 37
EXAMPLE 6 Dividing when the divisor is not linear
Divide x 3 - 3x 2 + 3x + 2 by x 2 + 1.
Solution Begin by writing x 2 + 1 as x 2 + 0x + 1.
x - 3 x 2 + 0x + 1�x 3 - 3x 2 + 3x + 2
x 3 + 0x 2 + x - 3x 2 + 2 x + 2 - 3x 2 + 0x - 3
2 x + 5
The quotient is x - 3 with remainder 2 x + 5. This result can also be written as
x - 3 + 2 x + 5 x 2 + 1
.
Now Try Exercise 41
5.6 Putting It All Together
Division by a Monomial Use the property
a + b d
= a
d +
b
d .
Be sure to divide the denominator into every term in the numerator.
2 x 3 + 4 x 2 x 2
= 2 x 3
2 x 2 +
4 x
2 x 2 = x +
2 x
and
a2 - 2a 4a
= a2
4a -
2a
4a =
a
4 -
1
2
CONCEPT EXPLANATION EXAMPLES
continued on next page
14 x x
= 14
14(x - 2) = 14 x - 28 Subtract: - 4 - ( - 28) = 24.
IS B
N 1-
25 6-
49 08
2- 2
Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.
CONCEPT EXPLANATION EXAMPLES
Division by a Polynomial
Is done similarly to the way long divi- sion of natural numbers is performed
If either the divisor or the dividend is missing a term, be sure to insert as a “place holder” the missing term with coefficient 0.
Divide x 2 + 3x + 3 by x + 1.
x + 2 x + 1�x 2 + 3x + 3
x 2 + x 2 x + 3 2 x + 2
1
The quotient is x + 2 with remainder 1, which can be expressed as
x + 2 + 1
x + 1 .
Checking a Result Dividend = (Divisor)(Quotient) + Remainder
When x 2 + 3x + 3 is divided by x + 1, the quotient is x + 2 with remainder 1. Thus
(x + 1)(x + 2) + 1 = x 2 + 3x + 3,
and the answer checks.
continued from previous page
CHAPTER 5 POLYNOMIALS AND EXPONENTS348
CONCEPTS AND VOCABULARY
1. a + bd = 2. a + b - c
d =
3. When dividing a polynomial by a monomial, the mono- mial must be divided into every of the polynomial.
4. (True or False?) The expressions 5x 2 + 2 x
2 x and 5x 2 + 1
are equal.
5. (True or False?) The expressions 5x 2 + 2 x
2 x and 5x 2 2 x are
equal.
6. Because 379 = 4 with remainder 1, it follows that 37 = # + .
7. Because 2 x 3 - x + 5 divided by x + 1 equals 2 x 2 - 2 x + 1 with remainder 4, it follows that 2 x 3 - x + 5 = _____ # _____ + _____.
8. When dividing 2 x 3 + 3x - 1 by x - 1, insert
into the dividend as a “place holder” for the missing x 2-term.
DIVISION BY A MONOMIAL
Exercises 9–16: Divide and check.
9. 6 x 2
3x 10.
- 5x 2
10x 4
11. z4 + z3
z 12.
t 3 - t t
13. a5 - 6a3
2a3 14.
b 4 - 4b 4b 2
15. y + 6y2
3y3 16.
8z2 - z 4z2
Exercises 17–26: Divide.
17. 4 x - 7x 4
x 2 18.
1 + 6 x 4
3x 3
19. 6y2 + 3y
3y3 20.
5z2 - 10z3
5z4
5.6 Exercises
21. 9x 4 - 3x + 6
3x 22.
y3 - 4y + 6 y
23. 12y4 - 3y2 + 6y
3y2 24.
2 x 2 - 6 x + 9 12 x
25. 15m4 - 10m3 + 20m2
5m2 26.
n8 - 8n6 + 4n4
2n5
IS B
N 1-256-49082-2
Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.
3495.6 DIVISION OF POLYNOMIALS
47. Thinking Generally If the quotient in a polynomial division problem is an integer, what must be true about the degrees of the dividend and divisor?
48. Thinking Generally If the quotient in a polynomial division problem is a polynomial of degree 1, what must be true about the degrees of the dividend and divisor?
APPLICATIONS
Exercises 49 and 50: Area of a Rectangle The area of a rectangle and its width are given. Find an expression for the length l.
49. 50.
51. Volume of a Box The volume V of a box is 2 x 3 + 4 x 2, and the area of its bottom is 2 x 2. Find the height of the box in terms of x. Make a possible sketch of the box, and label the length of each side.
52. Area of a Triangle A triangle has height h and area A = 2h 2 - 4h. Find its base b in terms of h. Make a possible sketch of the triangle, and label the height and base. (Hint: A = 12 bh.)
WRITING ABOUT MATHEMATICS
53. Suppose that one polynomial is divided into another polynomial and the remainder is 0. What does the product of the divisor and quotient equal? Explain.
54. A student simplifies the expression 4 x 3 - 1 4 x 2
to x - 1. Explain the student’s error.
2x A = 8x2
l l
x – 1 A = x2 – 1
27. 2 x 2 - 3x + 1
x - 2 28.
4 x 2 - x + 3 x + 2
Exercises 27–34: Divide and check.
29. x 2 + 2 x + 1
x + 1 30.
4 x 2 - 4 x + 1 2 x - 1
31. x 3 - x 2 + x - 2
x - 1 32.
2 x 3 + 3x 2 + 3x - 1 2 x + 1
33. x 3 + x 2 - 7x + 2
x - 2
34. x 3 + x 2 - 2 x + 12
x + 3
Exercises 35– 46: Divide.
35. 4 x 3 - 3x 2 + 7x + 3
4 x + 1
36. 10x 3 - x 2 - 17x - 7
5x + 2
37. x 3 - x + 2
x - 2
38. 6 x 3 + 8 x 2 + 4
3x + 4
39. (3x 3 + 2) , (x - 1)
40. ( - 3x 3 + 8 x 2 + x) , (3x + 4)
41. (x 3 + 3x 2 + 1) , (x 2 + 1)
42. (x 4 - x 3 + x 2 - x + 1) , (x 2 - 1)
43. x 3 + 1
x 2 - x + 1 44.
4 x 3 + 3x + 2 2 x 2 - x + 1
45. x 3 + 8 x + 2
46. x 4 - 16 x - 2
Checking Basic ConceptsSECTIONS5.5 and 5.6
1. Simplify each expression. Write the result with positive exponents.
(a) 9-2 (b) 3x -3
6 x 4 (c) (4ab -4)-2
2. Simplify each expression. Write the result with positive exponents.
(a) 1
z-5 (b)
x -3
y -6 (c) a 3
x 2 b-3
3. Write each number in scientific notation. (a) 45,000 (b) 0.000234 (c) 0.01
4. Write each expression in standard form. (a) 4.71 * 104 (b) 6 * 10-3
5. Simplify 25a 4 - 15a 3 5a 3
6. Divide 3x 2 - x - 4 by x - 1. State the quotient and remainder.
7. Divide x 4 + 2 x 3 - 2 x 2 - 5x - 2 by x 2 - 3. State the quotient and remainder.
8. Distance to the Sun The distance to the sun is
approximately 93 million miles. (a) Write this distance in scientific notation. (b) Light travels at 1.86 * 105 miles per second.
How long does it take for the sun’s light to reach Earth?
IS B
N 1-
25 6-
49 08
2- 2
Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.
350 CHAPTER 5 POLYNOMIALS AND EXPONENTS
CHAPTER 5 Summary SECTION 5.1 . RULES FOR EXPONENTS
Bases and Exponents The expression b n has base b and exponent n and equals the expression b # b # b # g # b, when n is a natural number.u
n times
Example: 23 has base 2 and exponent 3 and equals 2 # 2 # 2 = 8. Evaluating Expressions When evaluating expressions, evaluate exponents before performing addition, subtraction, multiplication, division, or negation. In general, operations within paren- theses should be evaluated before using the order of operations.
1. Evaluate exponents. 2. Perform negation. 3. Do multiplication and division from left to right. 4. Do addition and subtraction from left to right.
Example: - 32 + 3 # 4 = - 9 + 3 # 4 = - 9 + 12 = 3 Zero Exponents For any nonzero number b, b 0 = 1. Note that 00 is undefined.
Examples: 50 = 1 and ax y b0 = 1, where x and y are nonzero.
Product Rule For any real number a and natural numbers m and n,
am # an = am + n. Examples: 34 # 32 = 36 and x 3x 2x 4 = x 9 Power Rules For any real numbers a and b and natural numbers m and n,
(am)n = amn, (ab)n = anb n, and aa b bn = an
b n , b � 0.
Examples: (x 2)3 = x 6, (3x)4 = 34x 4 = 81x 4, and a2 y b3 = 23
y3 =
8
y3
SECTION 5.2 . ADDITION AND SUBTRACTION OF POLYNOMIALS
Terms Related to Polynomials Monomial A number, variable, or product of numbers and variables
raised to natural number powers
Degree of a Monomial Sum of the exponents of the variables
Coefficient of a Monomial The number in a monomial
Example: The monomial - 3x 2y3 has degree 5 and coefficient - 3.
Polynomial A monomial or the sum of two or more monomials
Term of a Polynomial Each monomial is a term of the polynomial.
Binomial A polynomial with two terms
Trinomial A polynomial with three terms
Degree of a Polynomial The degree of the term with highest degree
IS B
N 1-256-49082-2
Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.
351CHAPTER 5 SUMMARY
Opposite of a Polynomial The opposite is found by negating each term.
Example: 2 x 3 - 4 x + 5 is a trinomial with degree 3. Its opposite is - 2 x 3 + 4 x - 5.
Like Terms Two monomials with the same variables raised to the same powers
Examples: 3xy2 and - xy2 are like terms.
5x 3 and 3x 3 are like terms.
5x 2 and 5x are unlike terms.
Addition of Polynomials Combine like terms, using the distributive property.
Example: (2 x 2 - 4 x) + ( - x 2 - x) = (2 - 1)x 2 + ( - 4 - 1)x = x 2 - 5x
Subtraction of Polynomials Add the first polynomial to the opposite of the second polynomial.
Example: (4 x 4 - 5x) - (7x 4 + 6 x) = (4 x 4 - 5x) + ( - 7x 4 - 6 x) = (4 - 7)x 4 + ( - 5 - 6)x = - 3x 4 - 11x
SECTION 5.3 . MULTIPLICATION OF POLYNOMIALS
Multiplication of Monomials Use the commutative property and the product rule.
Examples: - 2 x 3 # 3x 2 = - 2 # 3 # x 3 # x 2 = - 6 x 5 (2 xy2)(3x 2y3) = 2 # 3 # x # x 2 # y2 # y3 = 6 x 3y5 c Assumed exponent of 1
Distributive Properties
a(b + c) = ab + ac and a(b - c) = ab - ac
Examples: 4 x(3x + 6) = 4 x # 3x + 4 x # 6 = 12 x 2 + 24 x ab(a2 - b 2) = ab # a2 - ab # b 2 = a3b - ab 3 Multiplication of Monomials and Polynomials Apply the distributive properties. Be sure to multiply every term in the polynomial by the monomial.
Example: - 2 x 2(4 x 2 - 5x - 3) = - 8 x 4 + 10x 3 + 6 x 2
Multiplication of Polynomials The product of two polynomials may be found by multiplying every term in the first polynomial by every term in the second polynomial. Be sure to combine like terms.
Examples: (x + 3)(2 x - 5) = 2 x 2 - 5x + 6 x - 15
= 2 x 2 + x - 15
(2 x + 1)(x 2 - 5x + 2) = 2 x 3 - 10x 2 + 4 x + x 2 - 5x + 2 = 2 x 3 - 9x 2 - x + 2
IS B
N 1-
25 6-
49 08
2- 2
Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.
352 CHAPTER 5 POLYNOMIALS AND EXPONENTS
SECTION 5.4 . SPECIAL PRODUCTS
Product of a Sum and Difference
(a + b)(a - b) = a2 - b 2
Examples: (x + 4)(x - 4) = x 2 - 16 (2r - 3t)(2r + 3t) = (2r)2 - (3t)2 = 4r 2 - 9t 2
Squaring Binomials
(a + b)2 = a2 + 2ab + b 2 and (a - b)2 = a2 - 2ab + b 2
Examples: (2 x + 1)2 = (2 x)2 + 2(2 x)1 + 12 = 4 x 2 + 4 x + 1 (z2 - 2)2 = (z2)2 - 2z2(2) + 22 = z4 - 4z2 + 4
Cubing Binomials To multiply (a + b)3, write it as (a + b)(a + b)2.
Example: (x + 4)3 = (x + 4)(x + 4)2
= (x + 4)(x 2 + 8 x + 16) Square the binomial. = x 3 + 8 x 2 + 16 x + 4 x 2 + 32 x + 64 Distributive property = x 3 + 12 x 2 + 48 x + 64 Combine like terms.
SECTION 5.5 . INTEGER EXPONENTS AND THE QUOTIENT RULE
Negative Integers as Exponents For any nonzero real number a and positive integer n,
a -n = 1
an .
Examples: 5-2 = 1
52 and x -4 =
1
x 4
The Quotient Rule For any nonzero real number a and integers m and n,
am
an = am - n.
Examples: 64
62 = 64 - 2 = 62 = 36 and
xy3
x 4y2 = x 1 - 4y3 - 2 = x -3y1 =
y
x 3
Other Rules For any nonzero real numbers a and b and positive integers m and n,
1
a -n = an,
a -n
b -m =
b m
an , and aa
b b-n = ab
a bn.
Examples: 1
4-3 = 43,
x -3
y -2 =
y2
x 3 , and a4
5 b-2 = a5
4 b2
Scientific Notation A real number a written as b * 10n, where 1 … 0b 0 6 10 and n is an integer Examples: 2.34 * 103 = 2340 Move the decimal point 3 places to the right. 2.34 * 10-3 = 0.00234 Move the decimal point 3 places to the left.
SECTION 5.6 . DIVISION OF POLYNOMIALS
Division of a Polynomial by a Monomial Divide the monomial into every term of the polynomial.
Example: 5x 3 - 10x 2 + 15x
5x =
5x 3
5x -
10x 2
5x +
15x
5x = x 2 - 2 x + 3
IS B
N 1-256-49082-2
Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.
353CHAPTER 5 REVIEW EXERCISES
Division of a Polynomial by a Polynomial Division of polynomials is performed similarly to long division of natural numbers.
Example: Divide 2 x 3 + 4 x 2 - 3x + 1 by x + 1.
2 x 2 + 2 x - 5 x + 1�2 x 3 + 4 x 2 - 3x + 1
2 x 3 + 2 x 2
2 x 2 - 3x 2 x 2 + 2 x
- 5x + 1 - 5x - 5
6
The quotient is 2 x 2 + 2 x - 5 with remainder 6, which can be written as
2 x 2 + 2 x - 5 + 6
x + 1 .
CHAPTER 5 Review Exercises SECTION 5.1
Exercises 1– 6: Evaluate the expression.
1. 53 2. - 34
3. 4( - 2)0 4. 3 + 32 - 30
5. - 52
5 6. a - 5
5 b2
Exercises 7–24: Simplify the expression.
7. 62 # 63 8. 105 # 107 9. z4 # z5 10. y2 # y # y3 11. 5x 2 # 6 x 7 12. (ab 3)(a3b) 13. (25)2 14. (m4)5
15. (ab)3 16. (x 2y3)4
17. (xy)3(x 2y4)2 18. (a2b 9)0
19. (r - t)4(r - t)5 20. (a + b)2(a + b)4
21. a 3 x - yb2 22. ax + y2 b3
23. 2 x 2(3x - 5) 24. 3x(4 x + x 3)
SECTION 5.2
Exercises 25 and 26: Identify the degree and coefficient of the monomial.
25. 6 x 7 26. - x 2y3
Exercises 27–30: Determine whether the expression is a polynomial. If it is, state how many terms and variables the polynomial contains. Then state its degree.
27. 8y 28. 8 x 3 - 3x 2 + x - 5
29. a2 + 2ab + b 2 30. 1 xy
31. Add the polynomials vertically.
3x 2 + 4 x + 8 2 x 2 - 5x - 5
32. Write the opposite of 6 x 2 - 3x - 7.
Exercises 33– 40: Simplify.
33. (4 x - 3) + ( - x + 7)
34. (3x 2 - 1) - (5x 2 + 12)
35. (x 2 + 5x + 6) - (3x 2 - 4 x + 1)
36. (x 2 + 3x - 5) + (2 x 2 - 5x - 1)
IS B
N 1-
25 6-
49 08
2- 2
Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.
354 CHAPTER 5 POLYNOMIALS AND EXPONENTS
37. (a3 + 4a2) + (a3 - 5a2 + 7a)
38. (4 x 3 - 2 x + 6) - (4x 3 - 6)
39. (xy + y2) + (4y2 - 4 xy)
40. (7x 2 + 2 xy + y2) - (7x 2 - 2 xy + y2)
SECTION 5.3
Exercises 41–54: Multiply and simplify.
41. - x 2 # x 3 42. - (r 2t 3)(rt) 43. - 3(2t - 5) 44. 2y(1 - 6y)
45. 6 x 3(3x 2 + 5x)
46. - x(x 2 - 2 x + 9)
47. - ab(a2 - 2ab + b 2)
48. (a - 2)(a + 5)
49. (8 x - 3)(x + 2) 50. (2 x - 1)(1 - x)
51. ( y2 + 1)(2y + 1) 52. ( y2 - 1)(2y2 + 1)
53. (z + 1)(z2 - z + 1)
54. (4z - 3)(z2 - 3z + 1)
Exercises 55 and 56: Multiply the expression (a) geometrically and (b) symbolically.
55. z(z + 1) 56. 2 x(x + 2)
SECTION 5.4
Exercises 57–72: Multiply.
57. (z + 2)(z - 2) 58. (5z - 9)(5z + 9) 59. (1 - 3y)(1 + 3y) 60. (5x + 4y)(5x - 4y)
61. (rt + 1)(rt - 1)
62. (2m2 - n2)(2m2 + n2)
63. (x + 1)2 64. (4 x + 3)2
65. ( y - 3)2 66. (2y - 5)2
67. (4 + a)2 68. (4 - a)2
69. (x 2 + y2)2
70. (xy - 2)2
71. (z + 5)3
72. (2z - 1)3
Exercises 73 and 74: Use the product of a sum and a dif- ference to evaluate the expression.
73. 59 # 61 74. 22 # 18 SECTION 5.5
Exercises 75–82: Simplify the expression.
75. 9-1 76. 3-2
77. 43 # 4-2 78. 10-6 # 103
79. 1
6-2 80.
57
59
81. (3-1 22)-2 82. (2-4 53)0 1
Exercises 83–98: Simplify the expression. Write the answer using positive exponents.
83. z-2 84. y -4
85. a -4 # a2 86. x 2 # x -5 # x 87. (2t)-2 88. (ab 2)-3
89. (xy)-2(x -2y)-1 90. x 6
x 2
91. 4 x
2 x 4 92.
20x 5y3
30xy6
93. aa b b5 94. 4
t -4
95. (3m3n)-2
(2m2n-3)3 96. a x -4y2
3xy -3 b-2
97. ax y b-2 98. a3u
2v b-1
Exercises 99–102: Write the expression in standard form.
99. 6 * 102 100. 5.24 * 104
101. 3.7 * 10-3 102. 6.234 * 10-2
Exercises 103–106: Write the number in scientific notation.
103. 10,000 104. 56,100,000
105. 0.000054 106. 0.001
Exercises 107 and 108: Evaluate the expression. Write the result in standard form.
107. (4 * 102)(6 * 104) 108. 8 * 103
4 * 104
SECTION 5.6
Exercises 109–116: Divide and check.
109. 5x 2 + 3x
3x 110.
6b 4 - 4b 2 + 2 2b 2
111. 3x 2 - x + 2
x - 1 112.
9x 2 - 6 x - 2 3x + 2
IS B
N 1-256-49082-2
Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.
355CHAPTER 5 REVIEW EXERCISES
114. 2 x 3 - x 2 - 1
2 x - 1 115.
x 3 - x 2 - x + 1 x 2 + 1
-
116. x 4 + 3x 3 + 8 x 2 + 7x + 5
x 2 + x + 1
113. 4 x 3 - 11x 2 - 7x - 1
4 x + 1
APPLICATIONS
117. Heart Rate An athlete starts running and continues for 10 seconds. The polynomial 12 t
2 + 60 calculates the heart rate of the athlete in beats per minute t sec- onds after beginning the run, where t … 10. (a) What is the athlete’s heart rate when the athlete
first starts to run? (b) What is the athlete’s heart rate after 10 seconds? (c) What happens to the athlete’s heart rate while
the athlete is running?
118. Areas of Rectangles Find a monomial equal to the sum of the areas of the rectangles. Calculate this sum for x = 3 feet and y = 4 feet.
2x
y
2x
y
2x
y
119. Area of a Rectangle Write a polynomial that gives the area of the rectangle. Calculate its area for z = 6 inches.
2
2
5
3
120. Area of a Square Find the area of the square whose sides have length x 2y.
x2y
x2y
121. Compound Interest If P dollars are deposited in an account that pays 6% annual interest, then the amount of money after 3 years is given by P(1 + 0.06)3. Find this amount when P = $700.
122. Volume of a Sphere The expression for the volume of a sphere with radius r is 43 pr
3. Find a polynomial that gives the volume of a sphere with radius x + 2. Leave your answer in terms of p.
x + 2
123. Height Reached by a Baseball A baseball is hit straight up. Its height h in feet above the ground after t seconds is given by t(96 - 16t). (a) Multiply this expression. (b) Evaluate both the expression in part (a) and the
given expression for t = 2. Interpret the result.
124. Rectangular Building A rectangular building has a perimeter of 1200 feet. (a) If one side of the building has length L, write
a polynomial expression that gives its area. (Be sure to multiply your expression.)
(b) Evaluate the expression in part (a) for L = 50 and interpret the answer.
125. Geometry Complete each part and verify that your answers are equal. (a) Find the area of the large square by multiplying
its length and width. (b) Find the sum of the areas of the smaller rect-
angles inside the large square.
5
x 5
x
126. Digital Picture A digital picture, including its border, is x + 4 pixels by x + 4 pixels, and the actual picture inside the border is x - 4 pixels by x - 4 pixels.
x + 4 x – 4
x + 4
x – 4
(a) Find a polynomial that gives the number of pix- els in the border.
(b) Let x = 100 and evaluate the polynomial.
IS B
N 1-
25 6-
49 08
2- 2
Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.
356 CHAPTER 5 POLYNOMIALS AND EXPONENTS
127. Federal Debt In 1990, the federal debt held by the public was $2.19 trillion, and the population of the United States was 249 million. Use scientific nota- tion to approximate the national debt per person. (Source: U.S. Department of the Treasury.)
128. Alcohol Consumption In 2007, about 239 million people in the United States were age 14 or older. They consumed, on average, 2.31 gallons of alcohol per person. Use scientific notation to estimate the total number of gallons of alcohol consumed by this age group. (Source: Department of Health and Human Services.)
8
1. Simplify each expression. (a) - 50 (b) - 92
2. Evaluate each expression by hand. (a) - 42 + 10 (b) 8-2
(c) 1
2-3 (d) - 3x 0
3. State how many terms and variables the polynomial 5x 2 - 3xy - 7y3 contains. Then state its degree.
4. Write the opposite of - x 3 + 4 x - 8.
Exercises 5–8: Simplify.
5. ( - 3x + 4) + (7x + 2)
6. ( y3 - 2y + 6) - (4y3 + 5)
7. (5x 2 - x + 3) - (4 x 2 - 2 x + 10)
8. (a3 + 5ab) + (3a3 - 3ab)
Exercises 9–16: Write the given expression with positive exponents.
9. 6y4 # 4y7 10. (a2b 3)2(ab 2) 11. x 7 # x -3 12. (a -1b 2)-3
13. ab(a2 - b 2) 14. a 3a2 2b -3
b-2 15.
12 xy4
6 x 2y 16. a 2
a + b b4
Exercises 17–22: Multiply and simplify.
24. Write 6.1 * 10-3 in standard form.
25. Write 5410 in scientific notation.
Exercises 26 and 27: Divide.
CHAPTER 5 Test Step-by-step test solutions are found on the Chapter Test Prep Videos available via the Video Resources on DVD, in , and on (search “RockswoldBeginAlg” and click on “Channels”).
17. 3x 2(4 x 3 - 6 x + 1)
19. (7y2 - 3)(7y2 + 3) 20. (3x - 2)2
18. (z - 3)(2z + 4)
21. (m + 3)3
22. ( y + 2)( y2 - 2y + 3)
23. Evaluate 78 # 82 using the product of a sum and a difference.
26. 9x 3 - 6 x 2 + 3x
3x 2 27.
x 3 + x 2 - x + 1 x + 2
28. Concert Tickets Tickets for a concert are sold for
$20 each. (a) Write a polynomial that gives the revenue from
selling t tickets. (b) Putting on the concert costs management $2000
to hire the band plus $2 for each ticket sold. What is the total cost of the concert if t tickets are sold?
(c) Subtract the polynomial that you found in part (b) from the polynomial that you found in part (a). What does this polynomial represent?
29. Areas of Rectangles Find a polynomial representing
the sum of the areas of two identical rectangles that have width 2 x and length 3x. Calculate this sum for x = 10 feet.
3x
2x
3x
2x
30. Volume of a Box Write a polynomial that repre- sents the volume of the box. Be sure to multiply your answer completely.
x + 6
x + 3 3x
31. Height Reached by a Golf Ball When a golf ball is hit into the air, its height in feet above the ground after t seconds is given by t(88 - 16t). (a) Multiply this expression. (b) Evaluate the expression in part (a) for t = 3.
Interpret the result.
IS B
N 1-256-49082-2
Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.
357CHAPTER 5 EXTENDED AND DISCOVERY EXERCISES
Exercises 1– 6: Arithmetic and Scientific Notation The product (4 * 103) * (2 * 102) can be evaluated as
(4 * 2) * (103 * 102) = 8 * 105,
and the quotient (4 * 103) , (2 * 102) can be evalu- ated as
4 * 103
2 * 102 =
4
2 *
103
102 = 2 * 101.
How would you evaluate (4 * 103) + (2 * 102)? How would you evaluate (4 * 103) - (2 * 102)? Make a con- jecture as to how numbers in scientific notation should be added and subtracted. Try your method on these prob- lems and then check your answers with a calculator set in scientific mode. Does your method work?
1. (4 * 103) + (3 * 103)
2. (5 * 10-2) - (2 * 10-2)
3. (1.2 * 104) - (3 * 103)
4. (2 * 102) + (6 * 101)
5. (2 * 10-1) + (4 * 10-2)
6. (2 * 10-3) - (5 * 10-2)
Exercises 7 and 8: Constructing a Box A box is con- structed from a rectangular piece of metal by cutting squares from the corners and folding up the sides. The square, cutout corners are x inches by x inches.
x
x x x
x x x
x
x
7. Suppose that the dimensions of the metal piece are 20 inches by 30 inches. (a) Write a polynomial that gives the volume of the
box. (b) Find the volume of the box for x = 4 inches.
8. Suppose that the metal piece is square with sides of length 25 inches. (a) Write a polynomial expression that gives the out-
side surface area of the box. (Assume that the box does not have a top.)
(b) Find this area for x = 3 inches.
Exercises 9–12: Calculators and Polynomials A graph- ing calculator can be used to help determine whether two polynomial expressions in one variable are equal. For example, suppose that a student believes that (x + 2)2 and x 2 + 4 are equal. Then the first two calculator tables shown demonstrate that the two expressions are not equal except for x = 0.
Y1�(X�2)2
X Y1 �3 1 �2 0 �1 1 0 4 1 9 2 16 3 25
Y1�X2�4
X Y1 �3 13 �2 8 �1 5 0 4 1 5 2 8 3 13
The next two calculator tables support the fact that (x + 1)2 and x 2 + 2 x + 1 are equal for all x.
Y1�(X�1)2
X Y1 �3 4 �2 1 �1 0 0 1 1 4 2 9 3 16
Y1�X2�2X�1
X Y1 �3 4 �2 1 �1 0 0 1 1 4 2 9 3 16
Use a graphing calculator to determine whether the first expression is equal to the second expression. If the expressions are not equal, multiply the first expression and simplify it.
9. 3x(4 - 5x), 12 x - 5x
10. (x - 1)2, x 2 - 1
11. (x - 1)(x 2 + x + 1), x 3 - 1
12. (x - 2)3, x 3 - 8
CHAPTER 5 Extended and Discovery Exercises
IS B
N 1-
25 6-
49 08
2- 2
Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.
358 CHAPTER 5 POLYNOMIALS AND EXPONENTS
CHAPTERS 1–5 Cumulative Review Exercises
Exercises 1 and 2: Evaluate each expression by hand.
1. (a) 18 - 2 # 5 (b) 42 , 7 + 2 2. (a) 21 - ( - 8) (b) - 73 , 1 - 149 2 Exercises 3 and 4: Solve the equation. Note that these equations may have no solutions, one solution, or infi- nitely many solutions.
3. (a) (x - 3) + x = 4 + x (b) 2(5x - 4) = 1 + 10x
4. (a) 2 + 6 x = 2(3x + 1) (b) 11x - 9 = - 31
5. Find the average speed of a car that travels 306 miles in 4 hours 30 minutes.
6. Write each value as a fraction in lowest terms. (a) 42% (b) 0.076
7. Graph the equation 4 x - 5y = 20.
8. Sketch a line with slope - 23 that passes through the point (1, 1).
9. Write the slope–intercept form for the line shown.
–2–3 –1 2 3
–2
–3
–1
1
3
x
y
10. Find the x-and y-intercepts for the graph of the equa- tion 2y = 3x - 6.
Exercises 11 and 12: Write the slope–intercept form of a line that satisfies the given information.
11. Parallel to 3x - 6y = 7, passing through (2, - 3) 12. Passing through ( - 2, - 5) and (1, 4)
Exercises 13–16: Solve the system of equations. Note that these systems may have no solutions, one solution, or infinitely many solutions.
Exercises 17 and 18: Shade the solution set for the system of inequalities.
13. 4 x + 3y = - 6 8 x + 6y = 12
14. x - 3y = 5 3x + y = 5
15. x + 4y = - 8 - 3x - 12y = 24
16. x - 5y = 30 2 x + y = - 6
17. x + y 6 3 y Ú x + 2
18. x - 2y 7 4 3x + y 6 6
19. Simplify the expression. (a) 3x 2 # 5x 3 (b) (x 3y)2(x 4y5)
20. Simplify. (a) (5x 2 - 3x + 4) - (3x 2 - 2 x + 1) (b) (7a3 - 4a2 - 5) + (5a3 + 4a2 + a)
21. Multiply and simplify. (a) (2 x + 3)(x - 7) (b) (y + 3)(y2 - 3y - 1)
(c) (4 x + 7)(4 x - 7) (d) (5a + 3)2
22. Simplify the expression. Write the answer using posi- tive exponents. (a) x -5 # x 3 # x (b) a 2
x 3 b-3
(c) 3x 2y -1
6 x -2y (d) (xy -2)3(x -2y)-2
23. Write 24,000,000,000 in scientific notation.
24. Write 4.71 * 10-7 in standard form.
25. Divide.
(a) 8 x 3 - 2 x
2 x (b)
2 x 2 + x - 14 x + 3
26. Price Decrease If the price of a computer is reduced from $1200 to $900, find the percent change.
27. Mixing an Acid Solution How many milliliters of a 3% acid solution should be added to 400 milliliters of a 6% acid solution to dilute it to a 5% acid solution?
28. Surface Area of a Box Use the drawing of the box to write a polynomial that represents the area of each of the following.
x + 5
x + 2 2x
(a) The bottom (b) The front (c) The right side (d) All six sides
IS B
N 1-256-49082-2
Beginning and Intermediate Algebra with Applications & Visualization, Third edition, by Gary K. Rockswold and Terry A. Krieger. Published by Addison Wesley. Copyright © 2013 by Pearson Education, Inc.
11
What Does the Healthcare
Manager Need to K n o w ?
2 C H A P T E R
HOW THE SYSTEM WORKS IN HEALTH CARE
The information that you, as a manager, work with is only one part of an overall system. To understand financial management, it is essential to recognize the overall sys- tem in which your organization operates. An order exists within the system, and it is generally up to you to find that order. Watch for how the information fits together. The four segments that make a healthcare financial sys- tem work are (1) the original records, (2) the informa- tion system, (3) the accounting system, (4) and the reporting system. Generally speaking, the original records provide evidence that some event has occurred; the information system gathers this evidence; the ac- counting system records the evidence, and the reporting system produces reports of the effect. The healthcare manager needs to know that these separate elements exist and that they work together for an end result.
THE INFORMATION FLOW
Structure of the Information System
Information systems can be simplistic or highly com- plex. They can be fully automated or semiautomated. Occasionally—even today—they can still be generated by hand and not by computer. (This last instance is be- coming rare and can happen today only in certain small and relatively isolated healthcare organizations that are not yet required to electronically submit their billings.)
We will examine a particular information system and point out the basics that a manager should be able to
After completing this chapter, you should be able to
1. Understand that four segments make a financial management system work.
2. Follow an information flow. 3. Recognize the basic system
elements. 4. Follow the annual management
cycle.
P r o g r e s s N o t e s
recognize. Figure 2-1 shows information system components for an ambulatory care setting. This complex system uses a clinical and financial data repository; in other words, both clin- ical and financial data are fed into the same system. An automated medical record is also linked to the system. These are basic facts that a manager should recognize about this am- bulatory information system.
In addition, the financial information, both outpatient and any relevant inpatient, is fed into the data repository. Scheduling-system data also enter the data repository, along with any relevant inpatient care plan and nursing information. Again, all of these are basic facts that a manager should recognize about this ambulatory care information system.
These items have all been inputs. One output from the clinical and financial data repos- itory (also shown in Figure 2-1) is insurance verification for patients through an electronic data information (EDI) link to insurance company databases. Insurance verification is daily operating information. Another output is decision-making information for managed care strategic planning, including support for demand, utilization, enrollment, and eligibility, plus some statistical support. The manager does not have to understand the specifics of all the inputs and outputs of this complex system, but he or she should recognize that these outputs occur when this ambulatory system is activated.
Function of Flowsheets
Flowsheets illustrate, as in this case, the flow of activities that capture information.1 Flow- sheets are useful because they portray who is responsible for what piece of information as it
12 CHAPTER 2 What Does the Healthcare Manager Need to Know?
Managed Care Systems –Enrollment/eligibility –Utilization management –Demand management –Algorithmic scheduling support
Insurance Verification
EDI Link
Enterprise-Wide Master Patient Index
Interface Engine
Clinical and Financial Data Repository
Practice Management System Scheduling
Ancillary Scheduling
Patient Accounting System Financial (OP and IP)
OR Scheduling
Other Provider Inpatient System
–Clinical order entry –IP care plans –IP nursing
Chart Tracking
(IP and OP)
A u t o m a t e d
M e d i c a l
R e c o r d
Figure 2–1 Information System Components for an Ambulatory Care Setting; OP, Outpatient; IP, Inpa- tient; OR, Operating Room.
The Information Flow 13
enters the system. The manager needs to realize the significance of such information. We give, as an example, obtaining confirmation of a patient’s correct address. The manager should know that a correct address for a patient is vital to the smooth operation of the sys- tem. An incorrect address will, for example, cause the billing to be rejected. Understand- ing this connection between deficient data (e.g., a bad address) and the consequences (the bill will be rejected by the payer and thus not be paid) illustrates the essence of good fi- nancial management knowledge.
We can examine two examples of patient information flows. The first, shown in Figure 2-2, is a physician’s office flowsheet for address confirmation. Four different personnel are involved, in addition to the patient. This physician has computed the cost of a bad address as $12.30 to track down each address correction. He pays close attention to the handling of this information because he knows there is a direct financial management consequence in his operation.
The second example, shown in Figure 2-3, is a health system flowsheet for verification of patient information. This flowsheet illustrates the process for a home care system. In this case, the flow begins not with a receptionist, as in the physician office example, but with a central database. This central database downloads the information and generates a sum- mary report to be reviewed the next day. Appropriate verification is then made in a series of steps, and any necessary corrections are made before the form goes to the billing depart- ment. The object of the flow is the same in both examples: that is, the billing must have a
Initiates Call
Mark Superbill if Change
Mark Superbill if Change
Patient
Enter Corrected Address in Computer
Copy Insurance
Card
Instruct Patient
to Correct
Financial Data
Check Address
Insurance
Review Patient
Records
Record Message
Receive Call
Phone Intake Entry/Exit
Receptionist
Ask Address Change
Ask Address Change
Medical Assistant
Ask Address Change
Doctor
Type Chart Label
Coder
Ask Insurance Change
Figure 2–2 Physician’s Office Flowsheet for Address Confirmation.
14 CHAPTER 2 What Does the Healthcare Manager Need to Know?
correct address to receive payment. But the flow is different within two different systems. A manager must understand how the system works to understand the consequences—then good financial management can prevail.
BASIC SYSTEM ELEMENTS
To understand financial management, it is essential to decipher the reports provided to the manager. To comprehend these reports, it is helpful to understand certain basic system el- ements that are used to create the information contained in the reports.
Chart of Accounts—The Map
The chart of accounts is a map. It outlines the elements of your company in an organized manner. The chart of accounts maps out account titles with a method of numeric coding. It is designed to compile financial data in a uniform manner that the user can decode.
The groupings of accounts in the chart of accounts should match the groupings of the organization. In other words, the classification on the organization chart (as discussed in the previous chapter) should be compatible with the groupings on the chart of accounts. Thus, if there is a human resources department on your facility’s organization chart, and if
Central intake enters demographics at time
referral received
Patient Accounts Clerk generates
Patient Information Summary next day
Data downloads to CDB overnight
Information on Patient Information Summary
verified by care manager at next
visit
Correct information written in by staff
Information Correct?
Form is placed in appropriate
care manager’s mailbox
Form turned in to Patient Accounts Representative
Data in CDB’s central intake updated
by Patient Accounts Representative
Form placed in billing folder
New labels generated if necessary
YesNo
Figure 2–3 Health System Flowsheet for Verification of Patient Information.
expenses are grouped by department in your facility, then we would expect to find a human resources grouping in the chart of accounts.
The manager who is working with financial data needs to be able to read and compre- hend how the dollars are laid out and how they are gathered together, or assembled. This assembly happens through the guidance of the chart of accounts. That is why we compare it to a map.
Basic guidance for healthcare charts of accounts is set out in publications such as that of Seawell’s Chart of Accounts for Hospitals.2 However, generic guides are just that—generic. Every organization exhibits differences in its own chart of accounts that express the unique aspects of its structure. We examine three examples to illustrate these differences. Remem- ber, we are spending time on the chart of accounts because your comprehension of de- tailed financial data may well depend on whether you can decipher your facility’s own chart of accounts mapping in the information forwarded for your use.
The first format, shown in Exhibit 2-1, is a basic use, probably for a smaller organization. The exhibit is in two horizontal segments, “Structure” and “Example.” There are three parts to the account number. The first part is one digit and indicates the financial statement element. Thus, our example shows “1,” which is for “Asset.” The second part is two digits and is the primary subclassification. Our example shows “10,” which stands for “Current Asset” in this case. The third and final part is also two digits and is the secondary subclassi- fication. Our example shows “11,” which stands for “Petty Cash—Front Office” in this case. On a report, this account number would probably appear as 1-10-11.
The second format, shown in Exhibit 2-2, is full use and would be for a large organiza- tion. The exhibit is again in two horizontal segments, “Structure” and “Example,” and there
Basic System Elements 15
Exhibit 2–1 Chart of Accounts, Format 1
Structure X XX XX Financial Primary Secondary Statement Subclassification Subclassification Element
Example 1 10 11 Asset Current Petty Cash—
Asset Front Office (Financial (Primary (Secondary Statement Subclassification) Subclassification) Element)
16 CHAPTER 2 What Does the Healthcare Manager Need to Know?
are now two line items appearing in the Example section. This full-use example has five parts to the account number. The first part is two digits and indicates the entity designator number. Thus, we conclude that there is more than one entity within this system. Our ex- ample shows “10,” which stands for “Hospital A.” The second part is two digits and indicates the fund designator number. Thus, we conclude that there is more than one fund within this system. Our example shows “10,” which stands for “General Fund.”
The third part of Exhibit 2-2 is one digit and indicates the financial statement element. Thus, the first line of our example shows “4,” which is for “Revenue,” and the second line of our example shows “6,” which is for “Expense.” (The third part of this example is the first part of the simpler example shown in Exhibit 2-1.) The fourth part is four digits and is the primary subclassification. Our example shows 3125, which stands for “Lab—Microbiology.” The number 3125 appears on both lines of this example, indicating that both the revenue and the expense belong to Lab—Microbiology. (The fourth part of this example is the sec- ond part of the simpler example shown in Exhibit 2-1. The simpler example used only two digits for this part, but this full-use example uses four digits.) The fifth and final part is two digits and is the secondary subclassification. Our example shows “03” on the first line, the revenue line, which stands for “Payer: XYZ HMO” and indicates the source of the revenue. On the second line, the expense line, our example shows “10,” which stands for “Clerical Salaries.” Therefore, we understand that these are the clerical salaries belonging to Lab—
Exhibit 2–2 Chart of Accounts, Format 2
Structure XX XX X XXXX XX Entity Fund Financial Primary Secondary Designator Designator Statement Subclassification Subclassification
Element
Example 10 10 4 3125 03 Hospital General Revenue Lab—Microbiology Payer: XYZ HMO A Fund
10 10 6 3125 10 Hospital General Expense Lab—Microbiology Clerical Salaries A Fund
(Entity (Fund (Financial (Primary (Secondary Designator) Designator) Statement Subclassification) Subclassification)
Element)
Basic System Elements 17
Microbiology in Hospital A. (The fifth part of this example is the third and final part of the simpler example shown in Exhibit 2-1.) On a report, these account numbers might appear as 10-10-4-3125-03 and 10-10-6-3125-10. Another optional use that is easier to read at a glance is 10104-3125-03 and 10106-3125-10.
Because every organization is unique and because the chart of accounts reflects that uniqueness, the third format, shown in Exhibit 2-3, illustrates a customized use of the chart of accounts. This example is adapted from a large hospital system. There are four parts to its chart of accounts number. The first part is an entity designator and designates a com- pany within the hospital system. The fund designator two-digit part, as traditionally used (see Exhibit 2-2), is missing here. The financial statement element one-digit part, as tradi- tionally used (see Exhibit 2-2), is also missing here. Instead, the second part of Exhibit 2-3 represents the primary classification, which is shown as an expense category (“Payroll”) in the example line. The third part of Exhibit 2-3 is the secondary subclassification, repre- senting a labor subaccount expense designation (“Regular per-Visit RN”). The fourth and final part of Exhibit 2-3 is another subclassification that indicates the department within the company (“Home Health”). On a report for this organization, therefore, the account num- ber 21-7000-2200-7151 would indicate the home care services company’s payroll for regu- lar per-visit registered nurses (RNs) in the home health department. Finally, remember that time spent understanding your own facility’s chart of accounts will be time well spent.
Exhibit 2–3 Chart of Accounts, Format 3
Structure XX XXXX XXXX XXXX Company Expense Subaccount Department
Category
(Entity (Primary (Secondary (Additional Designator) Classification) Subclassification) Subclassification)
Example 21 7000 2200 7151 Home Payroll Regular Home Health Care per-Visit RN Services
(Company) (Expense (Subaccount) (Department) Category)
18 CHAPTER 2 What Does the Healthcare Manager Need to Know?
Books and Records—Capture Transactions
The books and records of the financial information system for the organization ser ve to capture transactions. Figure 2-4 illustrates the relationship of the books and records to each other. As a single transaction occurs, the process begins. The individual transaction is recorded in the appropriate subsidiar y journal. Similar such transactions are then grouped and balanced within the subsidiary journal. At periodic intervals, the groups of transactions are gathered, summarized, and entered in the general ledger. Within the gen- eral ledger, the transaction groups are reviewed and adjusted. After such review and ad- justment, the transactions for the period within the general ledger are balanced. A document known as the trial balance is used for this purpose. The final step in the process is to create statements that reflect the transactions for the period. The trial balance is used to produce the statements.
All transactions for the period reside in the general ledger. The subsidiary journals are so named because they are “subsidiary” to the general ledger: in other words, they serve to support the general ledger. Figure 2-5 illustrates this concept. Another way to think of the subsidiary journals is to picture them as feeding the general ledger. The important point here is to understand the source and the flow of information as it is recorded.
Individual Transaction [begins process]
Individual Transaction Is Recorded
Similar Transactions Are Grouped and Balanced
Groups of Transactions Are Gathered and Summarized
Summarized Groups of Transactions Are Reviewed and Adjusted
Adjusted and Reviewed Transactions for Period Are Balanced
Statements Reflecting Transactions Are Created [ends process]
Creates Original Record
Into Subsidiary Journals
Within the Subsidiary Journals
Into the General Ledger
Within the General Ledger
Trial Balance Is Produced from the General Ledger for This Purpose
Statements Are Produced from the Trial Balance
Figure 2–4 The Progress of a Transaction. Source: Courtesy of Resource Group, Ltd., Dallas, Texas.
Reports—The Product
Reports are more fully treated in a subsequent chapter of this text (see Chapter 10). It is sufficient at this point to recognize that reports are the final product of a process that com- mences with an original transaction.
THE ANNUAL MANAGEMENT CYCLE
The annual management cycle affects the type and status of information that the manager is expected to use. Some operating information is “raw”—that is, unadjusted. When the same information has passed further through the system and has been verified, adjusted, and balanced, it will usually vary from the initial raw data. These differences are a part of the process just described.
Daily and Weekly Operating Reports
The daily and weekly operating reports generally contain raw data, as discussed in the pre- ceding paragraph. The purpose of such daily and weekly reports is to provide immediate operating information to use for day-by-day management purposes.
Quarterly Reports and Statistics
The quarterly reports and statistics generally have been verified, adjusted, and balanced. They are called interim reports because they have been generated some time during the reporting period of the organization and not at the end of that period. Managers often use quarterly reports as milestones. A common milestone is the quarterly budget review.
The Annual Management Cycle 19
Payroll Journal
Accounts Receivable
Journal
Cash Receipts Journal
Cash Disbursements
Journal
Accounts Payable Journal
GENERAL LEDGER
SUBSIDIARY JOURNALS
THE BOOKS
Figure 2–5 Recording Information: Relationship of Subsidiary Journals to the General Ledger. Source: Courtesy of Resource Group, Ltd., Dallas, Texas.
20 CHAPTER 2 What Does the Healthcare Manager Need to Know?
Annual Year-End Reports
Most organizations have a 12-month reporting period known as a fiscal year. A fiscal year, therefore, covers a period from the first day of a particular month (e.g., January 1) through the last day of a month that is one year, or 12 months, in the future (e.g., December 31). If we see a heading that reads, “For the year ended June 30,” we know that the fiscal year began on July 1 of the previous year. Anything less than a full 12-month year is called a “stub period” and is fully spelled out in the heading. If, therefore, a company is reporting for a three-month stub period ending on December 31, the heading on the report will read, “For the three-month period ended December 31.” An alternative treatment uses a heading that reads, “For the period October 1 to December 31.”
Annual year-end reports cover the full 12-month reporting period or the fiscal year. Such annual year-end reports are not primarily intended for managers’ use. Their primary pur- pose is for reporting the operations of the organization for the period to outsiders, or third parties.
Annual year-end reports represent the closing out of the information system for a spe- cific reporting period. The recording and reporting of operations will now begin a new cycle with a new year.
COMMUNICATING FINANCIAL INFORMATION TO OTHERS
The ability to communicate financial information effectively to others is a valuable skill. It is important to
• Create a report as your method of communication. • Use accepted terminology. • Use standard formats that are accepted in the accounting profession. • Begin with an executive summary. • Organize the body of the report in a logical flow. • Place extensive detail into an appendix.
The rest of this book will help you learn how to create such a report. Our book will also sharpen your communication skills by helping you better understand how heathcare fi- nance works.
INFORMATION CHECKPOINT
What Is Needed? An explanation of how the information flow works in your unit.
Where Is It Found? Probably with the information system staff; perhaps in the administrative offices.
How Is It Used? Study the flow and relate it to the paperwork that you handle.
KEY TERMS
Accounting System Chart of Accounts General Ledger Information System Original Records Reporting System Subsidiary Journals Trial Balance
DISCUSSION QUESTIONS
1. Have you ever been informed of the information flow in your unit or division? 2. If so, did you receive the information in a formal seminar or in an informal manner,
one-on-one with another individual? Do you think this was the best way? Why? 3. Do you know about the chart of accounts in your organization as it pertains to infor-
mation you receive? 4. If so, is it similar to one of the three formats illustrated in this chapter? If not, how is
it different? 5. Do you work with daily or weekly operating reports? With quarterly reports and sta-
tistics? 6. If so, do these reports give you useful information? How do you think they could be
improved?
Discussion Questions 21
P A R T
Healthcare Finance
O v e r v i e w
I
3
Introduction to Healthcare Finance 1
C H A P T E R
THE HISTORY
Financial management has a long and distinguished his- tory. Consider, for example, that Socrates wrote about the universal function of management in human en- deavors in 400 B.C. and that Plato developed the concept of specialization for efficiency in 350 B.C. Evidence of so- phisticated financial management exists for much earlier times: the Chinese produced a planning and control sys- tem in 1100 B.C., a minimum-wage system was developed by Hammurabi in 1800 B.C., and the Egyptians and Sumerians developed planning and record-keeping sys- tems in 4000 B.C.1
Many managers in early history discovered and redis- covered managerial principles while attempting to reach their goals. Because the idea of management thought as a discipline had not yet evolved, they formulated princi- ples of management because certain goals had to be ac- complished. As management thought became codified over time, however, the building of techniques for man- agement became more organized. Management as a dis- cipline for educational purposes began in the United States in 1881. In that year, Joseph Wharton created the Wharton School, offering college courses in business management at the University of Pennsylvania. It was the only such school until 1898, when the Universities of Chicago and California established their business schools. Thirteen years later, in 1911, 30 such schools were in operation in the United States.2
Over the long span of history, managers have all sought how to make organizations work more effectively. Financial management is a vital part of organizational
After completing this chapter, you should be able to
1. Discuss the three viewpoints of managers in organizations.
2. Identify the four elements of financial management.
3. Understand the differences between the two types of accounting.
4. Identify the types of organizations.
5. Understand the composition and purpose of an organization chart.
P r o g r e s s N o t e s
effectiveness. This book’s goal is to provide the keys to unlock the secrets of financial man- agement for nonfinancial managers.
THE CONCEPT
A Method of Getting Money in and out of the Business
One of our colleagues, a nurse, talks about the area of healthcare finance as “a method of getting money in and out of the business.” It is not a bad description. As we shall see, rev- enues represent inflow and expenses represent outflow. Thus, “getting money in” repre- sents the inflow (revenues), whereas “getting money out” (expenses) represents the outflow. The successful manager, through planning, organizing, controlling, and decision making, is able to adjust the inflow and outflow to achieve the most beneficial outcome for the organization.
HOW DOES FINANCE WORK IN THE HEALTHCARE BUSINESS?
The purpose of this book is to show how the various elements of finance fit together: in other words, how finance works in the healthcare business. The real key to understanding finance is understanding the various pieces and their relationship to each other. If you, the manager, truly see how the elements work, then they are yours. They become your tools to achieve management success.
The healthcare industry is a service industry. It is not in the business of manufacturing, say, widgets. Instead, its essential business is the delivery of healthcare services. It may have inventories of medical supplies and drugs, but those inventories are necessary to service de- livery, not to manufacturing functions. Because the business of health care is service, the explanations and illustrations within this book focus on the practice of financial manage- ment in the service industries.
VIEWPOINTS
The managers within a healthcare organization will generally have one of three views: (1) fi- nancial, (2) process, or (3) clinical. The way they manage will be influenced by which view they hold.
1. The financial view. These managers generally work with finance on a daily basis. The reporting function is part of their responsibility. They usually perform much of the strategic planning for the organization.
2. The process view. These managers generally work with the system of the organization. They may be responsible for data accumulation. They are often affiliated with the in- formation system hierarchy in the organization.
3. The clinical view. These managers generally are responsible for service delivery. They have direct interaction with the patients and are responsible for clinical outcomes of the organization.
4 CHAPTER 1 Introduction to Healthcare Finance
The Elements of Financial Management 5
Managers must, of necessity, interact with one another. Thus, managers holding differ- ent views will be required to work together. Their concerns will intersect to some degree, as illustrated by Figure 1-1. The nonfinancial manager who understands healthcare fi- nance will be able to interpret and negotiate successfully such interactions between and among viewpoints.
In summary, financial management is a discipline with a long and respected history. Healthcare service delivery is a business, and the concept of financial management assists in balancing the inflows and outflows that are a part of the business.
WHY MANAGE?
Business does not run itself. It requires a variety of management activities in order to oper- ate properly.
THE ELEMENTS OF FINANCIAL MANAGEMENT
There are four recognized elements of financial management: (1) planning, (2) control- ling, (3) organizing and directing, and (4) decision making. The four divisions are based on the purpose of each task. Some authorities stress only three elements (planning, con- trolling, and decision making) and consider organizing and directing as a part of the con- trolling element. This text recognizes organizing and directing as a separate element of financial management, primarily because such a large proportion of a manager’s time is taken up with performing these duties.
1. Planning. The financial manager identifies the steps that must be taken to accom- plish the organization’s objectives. Thus, the purpose is to identify objectives and then to identify the steps required for accomplishing these objectives.
2. Controlling. The financial manager makes sure that each area of the organization is following the plans that have been established. One way to do this is to study current reports and compare them with reports from earlier periods. This comparison often shows where the organization may need attention because that area is not effective. The reports that the manager uses for this purpose are often called feedback. The purpose of controlling is to ensure that plans are being followed.
3. Organizing and directing. When organizing, the financial manager decides how to use the resources of the organization to most effectively carry out the plans that have been established. When directing, the manager works on a day-to-day basis to keep the results of the organizing running efficiently. The purpose is to ensure effective re- source use and provide daily supervision.
Financial
Clinical
Process
Figure 1–1 3 Views of Mgmt within an Organization.
6 CHAPTER 1 Introduction to Healthcare Finance
4. Decision making. The financial manager makes choices among available alternatives. Decision making actually occurs parallel to planning, organizing, and controlling. All types of decision making rely on information, and the primary tasks are analysis and evaluation. Thus, the purpose is to make informed choices.
THE ORGANIZATION’S STRUCTURE
The structure of an organization is an important factor in management.
Organization Types
Organizations fall into one of two basic types: profit oriented or nonprofit oriented. In the United States, these designations follow the taxable status of the organizations. The profit- oriented entities, also known as proprietary organizations, are responsible for paying in- come taxes. Proprietary subgroups include individuals, partnerships, and corporations. The nonprofit organizations do not pay income taxes.
There are two subgroups of nonprofit entities: voluntary and government. Voluntary nonprofits have sought tax-exempt status. In general, voluntary nonprofits are associated with churches, private schools, or foundations. Government nonprofits, on the other hand, do not pay taxes because they are government entities. Government nonprofits can be (1) federal, (2) state, (3) county, (4) city, (5) a combination of city and county, (6) a hos-
pital taxing district (with the power to raise revenues through taxes), or (7) a state uni- versity (perhaps with a teaching hospital affiliated with the university). The organiza- tion’s type may affect its structure. Exhibit 1-1 summarizes the subgroups of both pro- prietary and nonprofit organizations.
Organization Charts
In a small organization, top management will be able to see what is happening. Exten- sive measures and indicators are not neces- sary because management can view overall operations. But in a large organization, top management must use the management control system to understand what is going on. In other words, to view operations, man- agement must use measures and indicators because he or she cannot get a firsthand overall picture of the total organization.
As a rule of thumb, an informal manage- ment control system is acceptable only if the
Exhibit 1–1 Types of Organizations
Profit Oriented—Proprietary Individual Partnership Corporation Other
Nonprofit—Voluntary Church Associated Private School Associated Foundation Associated Other
Nonprofit—Government Federal State County City City-County Hospital District State University Other
manager can stay in close contact with all aspects of the operation. Otherwise, a formal sys- tem is required. In the context of health care, therefore, a one-physician practice (Figure 1-2) could use an informal method, but a hospital system (Figure 1-3) must use a formal method of management control.
The structure of the organization will affect its financial management. Organization charts are often used to illustrate the structure of the organization. Each box on an orga- nization chart represents a particular area of management responsibility. The lines between the boxes are lines of authority.
In the health system organization chart illustrated in Figure 1-3, the president/chief ex- ecutive officer oversees seven senior vice presidents. Each senior vice president has vice presidents reporting to him or her in each particular area of responsibility designated on the chart. These vice presidents, in turn, have an array of other managers reporting to them at varying levels of managerial responsibility.
The organization chart also shows the degree of decentralization within the organiza- tion. Decentralization indicates the delegating of authority for decision making. The chart thus illustrates the pattern of how managers are allowed—or required—to make key deci- sions within the particular organization.
The purpose of an organization chart, then, is to indicate how responsibility is assigned to managers and to indicate the formal lines of communication and reporting.
TWO TYPES OF ACCOUNTING
Financial
Financial accounting is generally for outside, or third party, use. Thus, financial accounting emphasizes external reporting. External reporting to third parties in health care includes, for example, government entities (Medicare, Medicaid, and other government programs) and health plan payers. In addition, proprietary organizations may have to report to stock- holders, taxing district hospitals have to report to taxpayers, and so on.
Two Types of Accounting 7
Reception Scheduling
Billing Accounting
Front Office
Physician’s Assistant
Registered Nurse
Clinical Services
Physician
Figure 1–2 Physicians Office Organization Chart. Source: Courtesy of Resource Group, Ltd., Dallas, Texas.
8 CHAPTER 1 Introduction to Healthcare Finance M
e tr
o p
o lis
H e
a lth
S ys
te m
P re
si d
e n
t/ C
E O
S r.
V ic
e P
re si
d e
n t
S e
rv ic
e D
e liv
e ry
O p
e ra
tio n
s/ C
O O
In p
a tie
n t
O p
e ra
tio n
s
A m
b u
la to
ry O
p e
ra tio
n s
O th
e r
O p
e ra
tio n
s
O p
e ra
tio n
s T
Q I
S r.
V ic
e P
re si
d e
n t
H u
m a
n R
e so
u rc
e s
H u
m a
n R
e so
u rc
e s
O p
e ra
tio n
s
H R
P la
n n
in g
& P
la ce
m e
n t
L e
a rn
in g
S e
rv ic
e s
S r.
V ic
e P
re si
d e
n t
M e
d ic
a l M
a n
a g
e m
e n
t
P h ys
ic ia
n B
e n
e fit
s
P h ys
ic ia
n R
e cr
u itm
e n
t In
te g
ra tio
n
S r.
V ic
e P
re si
d e
n t
G e
n e
ra l C
o u
n se
l
L e
g a
l A
ff a
ir s
R is
k M
a n
a g
e m
e n
t
S r.
V ic
e P
re si
d e
n t
H u
m a
n A
ff a
ir s
C o
m m
u n
ity O
u tr
e a
ch
C o
m m
u n
ity H
e a
lth C
o u
n ci
l
C o
m m
u n
ity H
e a
lth I
m p
ro ve
m e
n t
P ro
g ra
m s
S r.
V ic
e P
re si
d e
n t
In fo
rm a
tio n
S ys
te m
s/ C
IO
In fo
S ys
te m
s O
p e
ra tio
n s
D a
ta M
a n
a g
e m
e n
t
In fo
S ys
te m
s D
e ve
lo p
m e
n t
S r.
V ic
e P
re si
d e
n t
F in
a n
ce /C
F O
C e
n tr
a l
B u si
n e ss
O ff ic
e
F in
a n
ce
In su
ra n
ce
M a
n a
g e
d C
a re
R e
a l E
st a
te F
a ci
lit ie
s/ D
e ve
lo p
m e
n t
P h ys
ic ia
n T
Q I
F ig
u re
1 –3
H ea
lt h
S ys
te m
O rg
an iz
at io
n C
h ar
t.
So u
rc e:
C o
u rt
es y
o f
R es
o u
rc e
G ro
u p
, L td
., D
al la
s, T
ex as
.
Key Terms 9
Financial reporting for external purposes must be in accordance with generally accepted accounting principles. Financial reporting is usually concerned with transactions that have already occurred: that is, it is retrospective.
Managerial
Managerial accounting is generally for inside, or internal, use. Managerial accounting, as its title implies, is used by managers. The planning and control of operations and related performance measures are common day-by-day uses of managerial accounting. Likewise, the reporting of profitability of services and the pricing of services are other common on- going uses of managerial accounting. Strategic planning and other intermediate and long- term decision making represent an additional use of managerial accounting.3
Managerial accounting intended for internal use is not bound by generally accepted ac- counting principles. Managerial accounting deals with transactions that have already oc- curred, but it is also concerned with the future, in the form of projecting outcomes and preparing budgets. Thus, managerial accounting is prospective as well as retrospective.
INFORMATION CHECKPOINT
What Is Needed? Reports for management purposes. Where Is It Found? With your supervisor. How Is It Used? To manage better. What Is Needed? Organization chart. Where Is It Found? With your supervisor or in the administrative offices. How Is It Used? To better understand the structure and lines of authority in
your organization.
KEY TERMS
Controlling Decision Making Financial Accounting Managerial Accounting Nonprofit Organization (also see Voluntary Organization) Organization Chart Organizing Planning Proprietary Organization (also see Profit-Oriented Organization)
10 CHAPTER 1 Introduction to Healthcare Finance
DISCUSSION QUESTIONS
1. What element of financial management do you perform most often in your job? 2. Do you perform all four elements? If not, why not? 3. Of the organization types described in this chapter, what type is the one you work
for? 4. Have you ever seen your company’s organization chart? If so, how decentralized is
it? 5. If you receive reports in the course of your work, do you believe that they are pre-
pared for outside (third party) use or for internal (management) use? What leads you to believe this?

Get help from top-rated tutors in any subject.
Efficiently complete your homework and academic assignments by getting help from the experts at homeworkarchive.com