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6
Rational Expressions
dvanced technical developments have made sports equipment
faster, lighter, and more responsive to the human body. Behind
the more flexible skis, lighter bats, and comfortable athletic
shoes lies the science of biomechanics, which is the study of
human movement and the factors that influence it.
Designing and testing an athletic shoe go hand in hand. While a shoe
is being designed, it is tested in a multitude of ways, including long-term
wear, rear foot stability, and strength of materials. Testing basketball
shoes usually includes an evaluation of the force applied to the ground
by the foot during running, jumping, and landing. Many biomechanics
laboratories have a special platform that can measure the force exerted when a player cuts from side to side as well as the force against
the bottom of the shoe. Force exerted in landing from a lay-up shot
can be as high as 14 times the weight of the body. Side-to-side
force is usually about 1 to 2 body weights in a cutting movement.
In Exercises 53 and 54 of Section 6.7 you will see how designers of athletic shoes use proportions to find the amount of
force on the foot and soles of shoes for activities such as
running and jumping.
A
Force (thousands of pounds)
5
4
3
2
1
0
50
100 150 200
Weight (pounds)
250
300
304
(6-2)
Chapter 6
Rational Expressions
6.1
Rational expressions in algebra are similar to the rational numbers in arithmetic. In
this section you will learn the basic ideas of rational expressions.
In this
section
●
Rational Expressions
●
Reducing to Lowest Terms
●
Reducing with the Quotient
Rule
●
REDUCING RATIONAL EXPRESSIONS
Dividing a b by b a
●
Factoring Out the Opposite
of a Common Factor
●
Writing Rational Expressions
Rational Expressions
A rational number is the ratio of two integers with the denominator not equal to 0.
For example,
3 9
, , 7, and 0
4 6
are rational numbers. A rational expression is the ratio of two polynomials with the
denominator not equal to 0. Because an integer is a monomial, a rational number is
a rational expression. The following expressions are rational expressions:
x2 1
,
x8
3a 2 5a 3
,
a9
3
,
7
w
We say that w is a rational expression because w can be written as w
.
1
A rational expression involving a variable has no value unless we assign a value
to the variable. Once the variable is given a value we can evaluate the expression.
We can discuss the value of a rational expression with or without the function notation introduced in Chapter 3.
E X A M P L E
1
calculator
Evaluating a rational expression
4x 1
a) Find the value of x 2 for x 3.
b) If R(x) close-up
To evaluate the rational expression in Example 1(a) with
a calculator, first use Y to
define the rational expression.
Be sure to enclose both numerator and denominator in
parentheses.
3x 2
,
2x 1
find R(4).
Solution
a) To find the value of
4x 1
x2
for x 3, replace x by 3 in the rational
expression:
4(3) 1 13
13
3 2
1
So the value of the rational expression is 13.
b) R(4) is the function notation for the value of the expression when x 4. To find
3x 2
R(4) replace x by 4 in R(x) :
2x 1
3(4) 2
R(4) 2(4) 1
Then find y1(3).
14
R(4) 2
7
So the value of the rational expression is 2 when x 4, or R(4) 2 (read “R of
4 is 2”).
■
Because the denominator cannot be zero, any number can be used in place of the
variable except numbers that cause the denominator to be zero.
6.1
E X A M P L E
2
Reducing Rational Expressions
(6-3)
305
Ruling out values for x
Which numbers cannot be used in place of x in each rational expression?
x2 1
x2
x5
b) c) a) x8
2x 1
x2 4
Solution
a) The denominator is 0 if x 8 0, or x 8. So 8 cannot be used in place
of x.
1
1
b) The denominator is zero if 2x 1 0, or x 2. So we cannot use 2 in
place of x.
c) The denominator is zero if x 2 4 0. Solve this equation:
x20
x2
x2 4 0
(x 2)(x 2) 0 Factor.
or
x 2 0 Zero factor property
or
x 2
So 2 and 2 cannot be used in place of x.
■
When dealing with rational expressions in this book, we will generally assume
that the variables represent numbers for which the denominator is not zero.
Reducing to Lowest Terms
Rational expressions are a generalization of rational numbers. The operations that
we perform on rational numbers can be performed on rational expressions in
exactly the same manner.
Each rational number can be written in infinitely many equivalent forms. For
example,
3
6
9
12 15
.
5 10 15 20 25
helpful
hint
Most students learn to convert 35 into 610 by dividing
5 into 10 to get 2 and then
multiply 2 by 3 to get 6. In algebra it is better to do this
conversion by multiplying the
numerator and denominator
of 35 by 2 as shown here.
Each equivalent form of 3 is obtained from 3 by multiplying both numerator and
5
5
denominator by the same nonzero number. This is equivalent to multiplying the
fraction by 1, which does not change its value. For example,
3 3
3 2
6
1
5 5
5 2 10
and
3 33
9
.
5 5 3 15
If we start with 6 and convert it into 3, we say that we are reducing 6 to lowest
10
5
10
terms. We reduce by dividing the numerator and denominator by the common
factor 2:
2 3 3
6
10 2 5 5
A rational number is expressed in lowest terms when the numerator and the denominator have no common factors other than 1.
CAUTION
We can reduce fractions only by dividing the numerator and
the denominator by a common factor. Although it is true that
24
6
,
10 2 8
306
(6-4)
Chapter 6
Rational Expressions
we cannot eliminate the 2’s, because they are not factors. Removing them from the
sums in the numerator and denominator would not result in 3.
5
Reducing Fractions
If a 0 and c 0, then
ab b
.
ac
c
To reduce rational expressions to lowest terms, we use exactly the same procedure as with fractions:
Reducing Rational Expressions
1. Factor the numerator and denominator completely.
2. Divide the numerator and denominator by the greatest common factor.
Dividing the numerator and denominator by the GCF is often referred to as
dividing out the GCF.
E X A M P L E
3
Reducing
Reduce to lowest terms.
30
a) 42
Solution
30 2 3 5
a) 42 2 3 7
5
7
Factor.
Divide out the GCF: 2 3 or 6.
x2 9
(x 3)(x 3)
b) 6x 18
6(x 3)
x3
6
study
tip
Find a clean, comfortable,
well-lit place to study. But
don’t get too comfortable.
Sitting at a desk is preferable
to lying in bed. Where you
study can influence your
concentration and your study
habits.
x2 9
b) 6x 18
Factor.
Divide out the GCF: x 3.
■
If two rational expressions are equivalent, then they have the same numerical
value for any replacement of the variables. Of course, the replacement must not give
us an undefined expression (0 in the denominator). So in Example 2(b) the equation
x2 9
x3
6x 18
6
is satisfied by all real numbers except x 3. The equation is an identity.
Reducing with the Quotient Rule
To reduce rational expressions involving exponential expressions, we use the
quotient rule for exponents from Chapter 4. We restate it here for reference.
6.1
Reducing Rational Expressions
(6-5)
307
Quotient Rule
Suppose a 0, and m and n are positive integers.
am
If m n, then n amn.
a
am
1
If m n, then n nm
.
a
a
E X A M P L E
4
Using the quotient rule in reducing
Reduce to lowest terms.
3a15
a) 7
6a
Solution
3a15
3a15
a) 7 7
6a
3 2a
a157
2
8
a
2
6x 4y 2
b) 4xy 5
Factor.
Quotient rule
6x 4y2 2 3x 4y2
b) 5
4xy5
2 2xy
3x 41
5
2y 2
3x 3
3
2y
Factor.
Quotient rule
■
The essential part of reducing is getting a complete factorization for the numerator and denominator. To get a complete factorization, you must use the techniques
for factoring from Chapter 5. If there are large integers in the numerator and
denominator, you can use the technique shown in Section 5.1 to get a prime factorization of each integer.
E X A M P L E
5
Reducing expressions involving large integers
Reduce
420
616
to lowest terms.
Solution
Use the method of Section 5.1 to get a prime factorization of 420 and 616:
7
535
31
05
22
10
Start here → 24
20
11
77
7
21
54
23
08
2616
308
(6-6)
Chapter 6
Rational Expressions
The complete factorization for 420 is 22 3 5 7, and the complete factorization
for 616 is 23 7 11. To reduce the fraction, we divide out the common factors:
420 22 3 5 7
616
23 7 11
35
2 11
15
22
■
Dividing a b by b a
In Section 4.2 you learned that a b (b a) 1(b a). So if a b is
divided by b a, the quotient is 1:
a b 1(b a)
ba
ba
1
We will use this fact in the next example.
E X A M P L E
6
Expressions with a b and b a
Reduce to lowest terms.
5x 5y
a) 4y 4x
Solution
5x 5y 5(x y)
a) 4y 4x 4( y x)
5
(1)
4
5
4
m2 n2
b) nm
Factor.
xy
1
yx
1
m2 n2 (m n)(m n)
b) nm
nm
1(m n)
m n
CAUTION
Factor.
mn
1
nm
We can reduce
ab
ba
■
ab
to 1, but we cannot reduce . There
is no factor that is common to the numerator and denominator of
ab
.
ab
ab
Factoring Out the Opposite of a Common Factor
If we can factor out a common factor, we can also factor out the opposite of that
common factor. For example, from 3x 6y we can factor out the common factor
6.1
Reducing Rational Expressions
(6-7)
309
3 or the common factor 3:
3x 6y 3(x 2y)
or
3x 6y 3(x 2y)
To reduce an expression, it is sometimes necessary to factor out the opposite of a
common factor.
E X A M P L E
7
Factoring out the opposite of a common factor
Reduce
3w 3w2
w2 1
to lowest terms.
Solution
We can factor 3w or 3w from the numerator. If we factor out 3w, we get a common factor in the numerator and denominator:
3w(1 w)
3w 3w 2
2
w 1
(w 1)(w 1)
3w
w1
3w
1w
Factor.
Since 1 w w 1, we divide out w + 1.
Multiply numerator and denominator by 1.
The last step in this reduction is not absolutely necessary, but we usually perform it
■
to make the answer look a little simpler.
The main points to remember for reducing rational expressions are summarized
in the following reducing strategy.
Strategy for Reducing Rational Expressions
1. Reducing is done by dividing out all common factors.
2. Factor the numerator and denominator completely to see the common
factors.
3. Use the quotient rule to reduce a ratio of two monomials.
4. You may have to factor out a common factor with a negative sign to get
identical factors in the numerator and denominator.
5. The quotient of a b and b a is 1.
Writing Rational Expressions
Rational expressions occur naturally in applications involving rates.
E X A M P L E
8
Writing rational expressions
Answer each question with a rational expression.
a) If a trucker drives 500 miles in x 1 hours, then what is his average speed?
b) If a wholesaler buys 100 pounds of shrimp for x dollars, then what is the price
per pound?
c) If a painter completes an entire house in 2x hours, then at what rate is she
painting?
310
(6-8)
Chapter 6
Rational Expressions
Solution
a) Because R D
, he is averaging
T
500
x1
mph.
x
100
dollars per pound
c) By completing 1 house in 2x hours, her rate is 1 house/hour.
■
b) At x dollars for 100 pounds, the wholesaler is paying
or x dollars/pound.
100
2x
WARM-UPS
True or false? Explain your answer.
1. A complete factorization of 3003 is 2 3 7 11 13. False
2. A complete factorization of 120 is 23 3 5. True
3. Any number can be used in place of x in the expression x2. True
5
4. We cannot replace x by 1 or 3 in the expression
5. The rational expression x2 reduces to x.
2
6.
7.
8.
9.
10.
6. 1
x1
.
x3
False
False
2x
x for any real number x. True
2
x13
1
20 7 for any nonzero value of x. True
x
x
a2 b2
reduced to lowest terms is a b. False
ab
If a b, then ab 1. False
ba
3x 6
The expression reduces to 3. True
x2
EXERCISES
Reading and Writing After reading this section, write out the
answers to these questions. Use complete sentences.
1. What is a rational number?
A rational number is a ratio of two integers with the denominator not 0.
2. What is a rational expression?
A rational expression is a ratio of two polynomials with the
denominator not 0.
3. How do you reduce a rational number to lowest terms?
A rational number is reduced to lowest terms by dividing
the numerator and denominator by the GCF.
4. How do you reduce a rational expression to lowest terms?
A rational expression is reduced to lowest terms by dividing the numerator and denominator by the GCF.
5. How is the quotient rule used in reducing rational expressions?
The quotient rule is used in reducing ratios of monomials.
6. What is the relationship between a b and b a?
The expressions a b and b a are opposites.
Evaluate each rational expression. See Example 1.
3x 3
7. Evaluate for x 2. 3
x5
3x 1
8. Evaluate for x 5. 1
4x 4
2x 9
9. If R(x) , find R(3). 5
x
20x 2
10. If R(x) , find R(1) 2
x8
x5
11. If R(x) , find R(2), R(4), R(3.02), and
x3
R(2.96). 0.6, 9, 401, 199
x 2 2x 3
12. If R(x) , find R(3), R(5), R(2.05),
x2
and R(1.999). 0, 4, 57.95, 3001.999
Which numbers cannot be used in place of the variable in each
rational expression? See Example 2.
x
3x
13. 1
14. 7
x1
x7
6.1
84
5
3
7a
16. 15. 3 2a 2
3a 5 3
2x 3
2y 1
4, 4
2, 3
18. 17. x2 16
y2 y 6
p1
m 31
19. 20. 2
5
Any number can be used.
Any number can be used.
Reduce each rational expression to lowest terms. Assume that
the variables represent only numbers for which the denominators are nonzero. See Example 3.
14 2
6 2
22. 21. 9
21 3
27
42 7
42 7
23. 24. 90 15
54 9
36a 2a
56y 7y
25. 26. 90
40 5
5
78 13
68 17
27. 28. 30w 5w
44y 11y
6x 2 3x 1
2w 2 w 1
29. 30. 6
3
2w
w
2x 4y
3m 9w
31. 32. 6y 3x
3m 6w
2
m 3w
3
m 2w
w2 49
a2 b2
33. 34. w7
ab
w7
ab
a2 1
x2 y2
35. 36. 2
2
a 2a 1
x 2xy y2
a1
xy
a1
xy
2x2 4x 2
2x2 10x 12
37. 38. 4x2 4
3x2 27
x1
2x 4
2x 2
3x 9
3x2 18x 27
x 3 3x 2 4x
39. 40. 21x 63
x 2 4x
x3
x1
7
Reduce each expression to lowest terms. Assume that all denominators are nonzero. See Example 4.
x10
y8
z3
41. 7
42. 5
43. 8
x
y
z
1
x3
y3
5
z
w9
4x 7
6y 3
44. 45. 5
46. 12
w
2x
3y 9
1
2
2x 2
3
w
y6
Reducing Rational Expressions
12m9n18
47. 6
8m n16
3m 3n 2
2
9x 20y
50. 6x 25y 3
3
52
2x y
9u9v19
48. 9
6u v14
3v 5
2
30a3bc
51. 18a7b17
5c
4
3a b16
(6-9)
6b10c4
49. 8b10c7
3
3
4c
15m10n3
52. 24m12np
5n2
8m2p
Reduce each expression to lowest terms. See Example 5.
210 35
616 14
53. 54. 264 44
660 15
936 3
231 11
56. 55. 624 2
168 8
5
21
8
630x
96y 2
57. 9 4
58. 5 3
108y 9y
300x 10x
924a23 33a4
270b75 18b63
59. 60. 19
448a
16
11
165b12
Reduce each expression to lowest terms. See Example 6.
3a 2b
5m 6n
61. 62. 2b 3a
6n 5m
1
1
2
2
2
r
s2
h t
63. 64. th
sr
h t
r s
2g 6h
5a 10b
65. 66. 2
9h2 g
4b2 a2
2
5
3h g
2b a
x2 x 6
1 a2
67. 68.
9 x2
a2 a 2
x 2
a1
x3
a2
Reduce each expression to lowest terms. See Example 7.
x 6
5x 20
69. 70. x6
3x 12
5
1
3
2y 6y2
y2 16
71. 72. 3 9y
8 2y
2y
4y
3
2
3x 6
8 4x
73. 74. 3x 6
8x 16
x2
x2
2x
2x 4
12a 6
2b2 6b 4
75. 76. 2
2a 7a 3
b2 1
6
2b 4
a3
b1
311
(6-10)
Chapter 6
Rational Expressions
Reduce each expression to lowest terms.
2x12
77. 8
4x
x4
2
4x2
78. 9
2x
2
7
x
2x 4
79. 4x
x2
2x
2x 4x2
80. 4x
1 2x
2
a4
81. 4a
2b 4
82. 2b 4
b2
b2
1
2c 4
83. 2
4c
2
c2
2t 4
84. 4 t2
2
t2
x2 4x 4
85. x2 4
x2
x2
3x 6
86. x2 4x 4
3
x2
2x 4
87. x2 5x 6
2
x3
2x 8
88. x2 2x 8
2
x2
2q8 q7
89. 2q6 q5
8s12
90. 6
12s 16s5
2s7
3s 4
q2
u2 6u 16
91. u2 16u 64
u2
u8
v2 3v 18
92. v2 12v 36
v3
v6
a3 8
93. 2a 4
a2 2a 4
2
4w2 12w 36
94. 2w3 54
2
w3
y3 2y2 4y 8
95. y2 4y 4
mx 3x my 3y
96. m2 3m 18
xy
m6
y2
Answer each question with a rational expression. Be sure to
include the units. See Example 8.
97. If Sergio drove 300 miles at x 10 miles per hour, then
how many hours did he drive?
300
hr
x 10
98. If Carrie walked 40 miles in x hours, then how fast did she
walk?
40
mph
x
99. If x 4 pounds of peaches cost $4.50, then what is the
cost per pound?
4.50
dollars/lb
x4
100. If nine pounds of pears cost x dollars, then what is the
price per pound?
x
dollars/lb
9
101. If Ayesha can clean the entire swimming pool in x hours,
then how much of the pool does she clean per hour?
1
pool/hr
x
102. If Ramon can mow the entire lawn in x 3 hours, then
how much of the lawn does he mow per hour?
1
lawn/hr
x3
Solve each problem.
103. Annual reports. The Crest Meat Company found that the
cost per report for printing x annual reports at Peppy Printing is given by the formula
150 0.60x
C(x) ,
x
where C(x) is in dollars.
a) Use the accompanying graph to estimate the cost per
report for printing 1000 reports.
b) Use the formula to find the cost per report for printing
1000, 5000, and 10,000 reports.
c) What happens to the cost per report as the number of
reports gets very large?
a) $0.75 b) $0.75, $0.63, $0.615
c) Approaches $0.60
Cost per report (dollars)
312
0.80
0.70
0.60
0.50
0.40
1
2
3
4
5
Number of reports (thousands)
FIGURE FOR EXERCISE 103
104. Toxic pollutants. The annual cost in dollars for removing
p% of the toxic chemicals from a town’s water supply is
given by the formula
500,000
C( p) .
100 p
Annual cost
(hundred thousand dollars)
6.2
a) Use the accompanying graph to estimate the cost for
removing 90% and 95% of the toxic chemicals.
b) Use the formula to determine the cost for removing
99.5% of the toxic chemicals.
c) What happens to the cost as the percentage of pollutants removed approaches 100%?
a) $50,000, $100,000
b) $1,000,000
c) Gets larger and larger without bound
6.2
In this
section
Multiplication and Division
(6-11)
313
5
4
3
2
1
0
90 91 92 93 94 95 96 97 98 99
Percentage of chemicals removed
FIGURE FOR EXERCISE 104
MULTIPLICATION AND DIVISION
In Section 6.1 you learned to reduce rational expressions in the same way that we
reduce rational numbers. In this section we will multiply and divide rational
expressions using the same procedures that we use for rational numbers.
●
Multiplication of Rational
Numbers
Multiplication of Rational Numbers
●
Multiplication of Rational
Expressions
Two rational numbers are multiplied by multiplying their numerators and multiplying their denominators.
●
Division of Rational
Numbers
●
Division of Rational
Expressions
●
Applications
Multiplication of Rational Numbers
If b 0 and d 0, then
a c ac
.
b d bd
E X A M P L E
1
Multiplying rational numbers
Find the product
helpful
hint
Did you know that the line
separating the numerator and
denominator in a fraction is
called the vinculum?
6 14
.
7 15
Solution
The product is found by multiplying the numerators and multiplying the denominators:
6 14
84
7 15 105
21 4
Factor the numerator and denominator.
21 5
4
Divide out the GCF 21.
5
The reducing that we did after multiplying is easier to do before multiplying. First
factor all terms, reduce, and then multiply:
6 14 2 3 2 7
7
3 5
7 15
4
5
■