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6
Rational
Expressions
nformation is everywhere—in the newspapers and magazines we
read, the televisions we watch, and the computers we use. And
now people are talking about the Information Superhighway,
which will deliver vast amounts of information directly to consumers’ homes. In the future the combination of telephone, television,
and computer will give us on-the-spot health care recommendations,
video conferences, home shopping, and perhaps even electronic voting
and driver’s license renewal, to name just a few. There is even talk of
500 television channels!
Some experts are concerned that the consumer will give up privacy
for this technology. Others worry about regulation, access, and content of the enormous international computer network.
Whatever the future of this technology, few people understand
how all their electronic devices work. However, this vast array of
electronics rests on physical principles, which are described by
mathematical formulas. In Exercises 49 and 50 of Section 6.6
we will see that the formula governing resistance for receivers connected in parallel involves rational expressions,
which are the subject of this chapter.
I
334
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Chapter 6
Rational Expressions
6.1
In this
section
●
Definition of Rational
Expressions
●
Domain
●
Reducing to Lowest Terms
●
Building Up the
Denominator
●
Rational Functions
●
Applications
PROPERTIES OF RATIONAL
EXPRESSIONS
A ratio of two integers is called a rational number; a ratio of two polynomials is
called a rational expression. Rational expressions are as fundamental to algebra as
rational numbers are to arithmetic. In this section we look carefully at some of the
properties of rational numbers and see how they extend to rational expressions.
Definition of Rational Expressions
A rational expression is the ratio of two polynomials with the denominator not
equal to zero. For example,
2
,
3
3a 5,
x3
,
2x 2 2
y2
,
5y
and
x2
x1
are rational expressions. The rational number 2 is a rational expression because 2
3
and 3 are monomials and 2 is a ratio of two monomials. If the denominator of a
3
rational expression is 1, it is usually omitted, as in the expression 3a 5.
Domain
helpful
hint
If the domain consists of all
real numbers except 5,
some people write R 5
for the domain. Even though
there are several ways to indicate the domain, you should
keep practicing interval notation because it is used in
algebra, trigonometry, and
calculus.
The domain of a rational expression is the set of all real numbers that can be used
in place of the variable. Because the denominator of a rational expression cannot
be zero, the domain of a rational expression consists of the set of real numbers
except those that cause the denominator to be zero. The domain of
x
x5
is the set of all real numbers excluding 5. In set-builder notation this set is
written as
x x 5,
and in interval notation it is written as
(, 5) (5, ).
E X A M P L E
1
Domain
Find the domain of each rational expression.
y2
x2
b) a) x9
5y
x3
c) 2x 2 2
Solution
a) The denominator is zero if x 9 0 or x 9. The domain is x x 9 or
(, 9) (9, ).
b) The denominator is zero if 5y 0 or y 0. The domain is y y 0 or
(, 0) (0, ).
6.1
Properties of Rational Expressions
(6-3)
335
c) The denominator is zero if 2x 2 2 0. Solve this equation.
2x 2 2 0
2(x 2 1) 0
2(x 1)(x 1) 0
x10
or
x10
x 1
or
x1
Factor out 2.
Factor completely.
Zero factor property
The domain is the set of all real numbers except 1 and 1. This set is written as
x x 1 and x 1, or in interval notation as
(, 1) (1, 1) (1, ).
■
CAUTION
The numbers that you find when you set the denominator
equal to zero and solve for x are not in the domain of the rational expression. The
solutions to that equation are excluded from the domain.
Reducing to Lowest Terms
Each rational number can be written in infinitely many equivalent forms. For
example,
8
2 4 6
10
. . . .
3 6 9 12 15
helpful
hint
Each equivalent form of 2 is obtained from 2 by multiplying both numerator and
3
3
denominator by the same nonzero number. For example,
Most students learn to convert 2 into 4 by dividing 3 into
3
2 2
2 2 4
1 3 3
3 2 6
6
6 to get 2 and then multiply 2
by 2 to get 4. In algebra it is
better to do this conversion
by multiplying the numerator
2
and denominator of 3 by 2 as
shown here.
and
2 2 3 6
.
3 3 3 9
Note that we are actually multiplying 2 by equivalent forms of 1, the multiplicative
3
identity.
If we start with 4 and convert it into 2, we are simplifying by reducing 4 to its
6
3
6
lowest terms. We can reduce as follows:
4 2 2 2
6 2 3 3
A rational number is expressed in its lowest terms when the numerator and denominator have no common factors other than 1. In reducing 4, we divide the numerator
6
and denominator by the common factor 2, or “divide out” the common factor 2. We
can multiply or divide both numerator and denominator of a rational number by the
same nonzero number without changing the value of the rational number. This fact
is called the basic principle of rational numbers.
Basic Principle of Rational Numbers
If
a
b
is a rational number and c is a nonzero real number, then
a ac
.
b bc
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Chapter 6
Rational Expressions
CAUTION
Although it is true that
5 23
,
6 24
we cannot divide out the 2’s in this expression because the 2’s are not factors. We
can divide out only common factors when reducing fractions.
Just as a rational number has infinitely many equivalent forms, a rational expression also has infinitely many equivalent forms. To reduce rational expressions
to its lowest terms, we follow exactly the same procedure as we do for rational
numbers: Factor the numerator and denominator completely, then divide out all
common factors.
E X A M P L E
helpful
2
hint
A negative sign in a fraction
can be placed in three locations:
1
1
1
2
2
2
The same goes for rational
expressions:
3x 2
3x 2
3x 2
5y
5y
5y
Reducing
Reduce each rational expression to its lowest terms.
18
2a7b
a) b) 2
42
a b3
Solution
a) Factor 18 as 2 32 and 42 as 2 3 7:
2 32
18
42 2 3 7
3
7
Factor.
Divide out the common factors.
b) Because this expression is already factored, we use the quotient rule for exponents to reduce:
2a7b 2a5
2
a b3
b2
■
In the next example we use the techniques for factoring polynomials that we
learned in Chapter 5.
E X A M P L E
3
Reducing
Reduce each rational expression to its lowest terms.
w2
2x 2 18
2a 3 16
a) b) c) 2
2
x x6
16 4a
2w
Solution
2x 2 18
2(x 2 9)
a) 2
x x 6 (x 2)(x 3)
2(x 3)(x 3)
(x 2)(x 3)
2x 6
x2
Factor.
Factor completely.
Divide out the common factors.
6.1
Properties of Rational Expressions
(6-5)
337
b) Factor out 1 from the numerator to get a common factor:
w 2 1(2 w)
1
2w
(2 w)
2a 3 16
2(a 3 8)
c) 2 2
16 4a
4(a 4)
study
tip
Studying in a quiet place is
better than studying in a noisy
place. There are very few people who can listen to music or
a conversation and study at
the same time.
Factoring out 4 will give the
common factor a 2.
2(a 2)(a2 2a 4)
2 2(a 2)(a 2)
Difference of two cubes,
difference of two squares
a2 2a 4
2a 4
Divide out common factors.
■
The rational expressions in Example 3(a) are equivalent because they have the
same value for any replacement of the variables, provided that the replacement is in
the domain of both expressions. In other words, the equation
2x 6
2x 2 18
2
x2
x x6
is an identity. It is true for any value of x except 2 and 3.
The main points to remember for reducing rational expressions are summarized
as follows.
helpful
hint
Since 1(a b) b a,
placement of a negative
sign in a rational expression
changes the appearance of
the expression:
3 x (3 x)
x2
x2
x3
x2
3x
3x
x 2 (x 2)
3x
2x
Strategy for Reducing Rational Expressions
1. All reducing is done by dividing out 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 involving
exponents.
4. We may have to factor out a common factor with a negative sign to get
identical factors in the numerator and denominator.
Building Up the Denominator
In Section 6.3 we will see that only rational expressions with identical denominators can be added or subtracted. Fractions without identical denominators can be
converted to equivalent fractions with a common denominator by reversing the
procedure for reducing fractions to its lowest terms. This procedure is called
building up the denominator.
Consider converting the fraction 1 into an equivalent fraction with a denomina3
tor of 51. Any fraction that is equivalent to 1 can be obtained by multiplying the nu3
merator and denominator of 1 by the same nonzero number. Because 51 3 17,
3
we multiply the numerator and denominator of 1 by 17 to get an equivalent fraction
3
with a denominator of 51:
1 1
1 17 17
1 3 3
3 17 51
338
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Chapter 6
E X A M P L E
helpful
4
hint
Notice that reducing and
building up are exactly the
opposite of each other. In reducing you remove a factor
that is common to the numerator and denominator, and in
building-up you put a common factor into the numerator and denominator.
Rational Expressions
Building up the denominator
Convert each rational expression into an equivalent rational expression that has the
indicated denominator.
2 ?
5
?
b) ,
a) , 34
2
7 42
3a b 9a b
Solution
a) Factor 42 as 42 2 3 7, then multiply the numerator and denominator of 2
7
by the missing factors, 2 and 3:
2 2 2 3 12
7 7 2 3 42
b) Because 9a 3b 4 3ab3 3a 2b, we multiply the numerator and denominator by
3ab 3:
5
5 3ab 3
2 3a b 3a 2b 3ab 3
15ab3
34
9a b
■
When building up a denominator to match a more complicated denominator, we
factor both denominators completely to see which factors are missing from the simpler denominator. Then we multiply the numerator and denominator of the simpler
expression by the missing factors.
E X A M P L E
helpful
5
hint
Multiplying the numerator
and denominator of a rational
expression by 1 changes the
appearance of the expression:
6 x 1(6 x)
x 7 1(x 7)
x6
7x
y5
1( y 5)
4 y 1(4 y)
5y
4y
Building up the denominator
Convert each rational expression into an equivalent rational expression that has the
indicated denominator.
5
?
x2
?
b) , a) , 2
2a 2b 6b 6a
x 3 x 7x 12
Solution
a) Factor both 2a 2b and 6b 6a to see which factor is missing in 2a 2b.
Note that we factor out 6 from 6b 6a to get the factor a b :
2a 2b 2(a b)
6b 6a 6(a b) 3 2(a b)
Now multiply the numerator and denominator by the missing factor, 3:
5
5(3)
15
2a 2b (2a 2b)(3) 6b 6a
b) Because x 2 7x 12 (x 3)(x 4), multiply the numerator and denominator by x 4:
x 2 (x 2)(x 4)
x 2 6x 8
x 3 (x 3)(x 4) x 2 7x 12
■
Rational Functions
A rational expression can be used to determine the value of a variable. For example, if
3x 1
y
,
x2 4
6.1
Properties of Rational Expressions
(6-7)
339
then we say that y is a rational function of x. We can also use function notation as
shown in the next example.
E X A M P L E
6
Evaluating a rational function
Find R(3), R(1), and R(2) for the rational function
3x 1
R(x) .
x2 4
calculator
Solution
To find R(3), replace x by 3 in the formula:
331 8
R(3) 32 4
5
close-up
To check, use Y= to enter
y1 (3x 1)(x 2 4).
Then use the variables feature
(VARS) to find y1(3) and y1(1).
To find R(1), replace x by 1 in the formula:
3(1) 1
R(1) (1)2 4
4 4
3 3
We cannot find R(2) because 2 is not in the domain of the rational expression.
■
Applications
A rational expression can occur in finding an average cost. The average cost of
making a product is the total cost divided by the number of products made.
E X A M P L E
7
Average cost function
Mercedes Benz spent $700 million to develop its new 1999 M class SUV, which
will sell for around $40,000 (Motor Trend, July 1998, www.motortrend.com). If the
cost of manufacturing the SUV is $30,000 each, then what rational function gives
the average cost of developing and manufacturing x vehicles? Compare the average
cost per vehicle for manufacturing levels of 10,000 vehicles and 100,000 vehicles.
Solution
The polynomial 30,000x 700,000,000 gives the cost in dollars of developing and
manufacturing x vehicles. The average cost per vehicle is given by the rational
function
30,000x 700,000,000
AC(x) .
x
If x 10,000, then
30,000(10,000) 700,000,000
AC(10,000) 100,000.
10,000
If x 100,000, then
30,000(100,000) 700,000,000
AC(100,000) 37,000.
100,000
The average cost per vehicle when 10,000 vehicles are made is $100,000, whereas
■
the average cost per vehicle when 100,000 vehicles are made is $37,000.
340
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Chapter 6
WARM-UPS
Rational Expressions
True or false? Explain.
1. A rational number is a rational expression. True
2. The expression 2x is a rational expression. True
x1
3. The domain of the rational expression 3 is 2. False
x2
1
2x 5
4. The domain of is x x 9 and x . True
2
(x 9)(2x 1)
5. The domain of x1 is (, 2) (2, 1) (1, ). False
x2
2
x2
6. The rational expression 5x
reduces to .
15
3
False
2
x
7. Multiplying the numerator and denominator of x by x yields .
x1
x2 1
False
2
8. The expression 2 is equivalent to . True
3x
x3
3
2
9. The equation 4x 2x is an identity.
6x
3
True
y
reduced to its lowest terms is x y.
10. The expression x
2
2
xy
6. 1
False
EXERCISES
Reading and Writing After reading this section, write out the
answers to these questions. Use complete sentences.
1. What is a rational expression?
A rational expression is a ratio of two polynomials with the
denominator not equal to zero.
2. What is the domain of a rational expression?
The domain of a rational expression is all real numbers
except those that cause the denominator to be zero.
3. What is the basic principle of rational numbers?
The basic principle of rational numbers says that
(ab)(ac) bc, provided a and c are not zero.
4. How do we reduce a rational expression to lowest terms?
To reduce a rational expression, factor the numerator and
denominator completely and then divide out the common
factors.
5. How do you build up the denominator of a rational
expression?
We build up the denominator by multiplying the numerator
and denominator by the same expression.
6. What is average cost?
Average cost is total cost divided by the number of items.
Find the domain of each rational expression. See Example 1.
3x
7. x x 1
x1
x
8. x x 5
x5
2z 5
9. z z 0
7z
z 12
10. z z 0
4z
5y 1
11. y2 4
y y 2 and y 2
2y 1
12. y2 9
y y 3 and y 3
2a 3
13. a 2 5a 6
a a 2 and a 3
3b 1
14. 2b2 7b 4
b b 2 and b 4
x1
15. x 2 4x
2x
16. 2
3x 9x
1
x x 4 and x 0
x x 3 and x 0
x1
17. x x 3 and x 0 and x 2
x 3 x 2 6x
x 2 3x 4
18. x x 1 and x 0 and x 1
2x 5 2x
Reduce each rational expression to its lowest terms. See Examples 2 and 3.
6
42
14
20. 21. 19. 57
210
91
1
2
2
5
19
13
242
22. 154
11
7
2x 2
23. 4
x1
2
3a 3
24. 3
a1
6.1
3x 6y
25. 10y 5x
3
5
36y 3z8
28. 54y2z9
2y
3z
a3b2
31. 3
a a4
b2
1a
ab
33. 2b 2a
1
2
3x 6
35. 3x
x2
x
a3 b3
37. ab
39.
41.
43.
45.
5b 10a
26. 2a b
5
ab 2
27. 3
ab
b
2
a
6a 3b12c5
30. 8ab4c9
3a2b8
4c4
2w 2x 3y
29. 6wx 5y2
w
3x2y
b8 ab5
32. ab5
3
b a
a
2m 2n
34. 4n 4m
1
2
7x 14
36. 7x
x2
x
27x 3 y3
38. 6x 2y
2
9x
3xy y2
a2 ab b2
2
4x 2 4
2a2 2b2
40. 4x 2 4
2a2 2b2
2
2
x 1
a b2
x2 1
a2 b2
2x 2 2x 12
2x 2 10x 12
42. 2
4x 36
2x 2 8
x3
x2
x2
2x 6
x 3 7x 2 4x
2x 4 32
44. x 3 16x
4x 8
2
x 7x 4
(x 2)(x 2 4)
x 2 16
2
ab 3a by 3y
2x 2 5x 3
46. a2 y2
2x 2 11x 5
b3
x3
ay
x5
Convert each rational expression into an equivalent rational
expression that has the indicated denominator. See Examples 4
and 5.
1 ?
2 ?
47. , 48. , 5 50
3 9
10
6
50
9
1 ?
3
?
49. , 2
50. ,
35
2
x 3x
ab a b
3x
3a2b3
2
3
3x
a b5
Properties of Rational Expressions
(6-9)
341
5
?
51. , x 1 x 2 2x 1
5x 5
x 2 2x 1
1
?
53. , 2x 2 6x 6
3
6x 6
x
?
52. , x 3 x2 9
2
x 3x
x2 9
2
?
54. , 3x 4 15x 20
10
15x 20
?
55. 5, a
5a
a
?
56. 3, a1
3a 3
a1
x2
?
57. , 2
x 3 x 2x 3
2
x x2
x 2 2x 3
x
?
58. , 2
x 5 x x 20
2
x 4x
2
x x 20
7
?
59. , x1 1x
7
1x
1
?
60. , a b 2b 2a
2
2b 2a
3
?
61. , x 2 x3 8
2
3x 6x 12
x3 8
x
?
62. , x 2 x3 8
3
2
x 2x 4x
x3 8
x2
?
63. , 3x 1 6x2 13x 5
2x 2 9x 10
6x 2 13x 5
a
?
64. , 2a 1 4a2 16a 9
2a2 9a
2
4a 16a 9
Find the indicated value for each given rational expression. See
Example 6.
3x 5
4
65. R(x) , R(3) x4
7
5x
66. T(x) , T(9) 1
x5
y2 5
1
67. H(y) , H(2) 3y 4
10
3 5a
22
68. G(a) , G(5) 2a 7
17
4b3 1
69. W(b) , W(2) Undefined
b2 b 6
x3
70. N(x) , N(3) Undefined
x 3 2x 2 2x 3
In place of each question mark in Exercises 71–90, put an
expression that will make the rational expressions equivalent.
?
?
1
72. 4 71. 3
3 21
7
12
21
3
(6-10)
10
73. 5 ?
10
2
?
3
75. 2
a a
3a
2
a
2
?
77. ab ba
2
ba
2
?
79. x 1 x2 1
2x 2
x2 1
2
2
81. w3
?
2
3w
2x 4 ?
83. 3
6
x2
3
x4
1
85. x 2 16 ?
1
x4
3a 3 ?
87. 3a
a
a1
a
1
?
89. x 1 x3 1
x2 x 1
x3 1
Chapter 6
Rational Expressions
3 12
74. 4
?
12
16
5 10
76. y
?
10
2y
3
?
78. x4 4x
3
4x
5
?
80. x 3 x2 9
5x 15
x2 9
2
2
82. 5x ?
2
x5
2x 3 1
84. 4x 6 ?
1
2
2x 2 x 1
86. 2x
?
x1
x
x3
1
88. 2
x 9 ?
1
x3
x 2 2x 4
?
90. x2
x2 4
x3 8
x2 4
Reduce each rational expression to its lowest terms. Variables
used in exponents represent integers.
x 2a 4
x 2b 3x b 18
92. 91. a
x 2
x 2b 36
xb 3
xa 2
xb 6
x a m wx a wm
x 3a 8
93. 94. 2a
2
2a
x m
x 2x a 4
1w
xa 2
xa m
2x 2a1 3x a1 x
x 3b1 x
95. 96.
4x 2a1 x
x 2b1 x
x 2b x b 1
xb 1
xa 1
2x a 1
Solve each problem. See Example 7.
97. Driving speed. If Jeremy drives 500 miles in 2x hours,
then what rational expression represents his speed in
miles per hour (mph)?
250
mph
x
98. Filing suit. If Marsha files 48 suits in 2x 2 work days,
then what rational expression represents the rate (in suits
per day) at which she is filing suits?
24
suits per day
x1
99. Wedding bells. Wheeler Printing Co. charges $45 plus
$0.50 per invitation to print wedding invitations.
a) Write a rational function that gives the average cost in
dollars per invitation for printing n invitations.
b) How much less does it cost per invitation to print 300
invitations rather than 200 invitations?
c) As the number of invitations increases, does the average cost per invitation increase or decrease?
d) As the number of invitations increases, does the total
cost of the invitations increase or decrease?
0.50n 45
a) A(n) dollars
n
b) 7.5 cents c) decreases d) increases
2
Cost per invitation
(in dollars)
342
1
0
0
100
200
300
Number of invitations
FIGURE FOR EXERCISE 99
100. Rose Bowl bound. A travel agent offers a Rose Bowl
package including hotel, tickets, and transportation. It
costs the travel agent $50,000 plus $300 per person to
charter the airplane. Find a rational function that gives the
average cost in dollars per person for the charter flight.
How much lower is the average cost per person when
200 people go compared to 100 people?
50,000 300n
A(n) dollars, $250 per person
n
101. Solid waste recovery. The amount of municipal solid
waste generated in the United States in the year 1960 n
is given by the polynomial
3.43n 87.24,
whereas the amount recycled is given by the polynomial
0.053n2 0.64n 6.71,
6.2
section
●
●
●
343
103. Exploration. Use a calculator to find R(2), R(30),
R(500), R(9,000), and R(80,000) for the rational
expression
x3
R(x) .
2x 1
Round answers to four decimal places. What can you
conclude about the value of R(x) as x gets larger and
larger without bound?
The value of R(x) gets closer and closer to 1.
102. Higher education. The total number of degrees awarded
in U.S. higher education in the year 1990 n is given in
thousands by the polynomial 41.7n 1429, whereas the
number of bachelor’s degrees awarded is given in thousands by the polynomial 25.2n 1069 (National Center
for Education Statistics, www.nces.ed.gov).
a) Write a rational function p(n) that gives the percentage
of bachelor’s degrees among the total number of
degrees conferred for the year 1990 n.
b) What percentage of the degrees awarded in 2010 will
be bachelor’s degrees?
25.2n 1069
a) p(n) b) 69.5%
41.7n 1429
In this
(6-11)
GET TING MORE INVOLVED
where the amounts are in millions of tons (U.S. Environmental Protection Agency, www.epa.gov).
a) Write a rational function p(n) that gives the fraction of
solid waste that is recovered in the year 1960 n.
b) Find p(0), p(30), and p(50).
0.053n2 0.64n 6.71
a) p(n) 3.43n 87.24
b) 7.7%, 18.5%, 41.4%
6.2
Multiplication and Division
2
104. Exploration. Use a calculator to find H(1,000),
H(100,000), H(1,000,000), and H(10,000,000) for the
rational expression
7x 50
H(x) .
3x 91
Round answers to four decimal places. What can you
conclude about the value of H(x) as x gets larger and
larger without bound?
7
The value of H(x) gets closer and closer to .
3
MULTIPLICATION AND DIVISION
In Chapter 5 you learned to add, subtract, multiply, and divide polynomials. In this
chapter you will learn to perform the same operations with rational expressions. We
begin in this section with multiplication and division.
Multiplying Rational
Expressions
Multiplying Rational Expressions
Dividing a b by b a
We multiply two rational numbers by multiplying their numerators and multiplying
their denominators. For example,
Dividing Rational
Expressions
6
7
84
14
21 4 4
.
15 105 21 5 5
Instead of reducing the rational number after multiplying, it is often easier to reduce
before multiplying. We first factor all terms, then divide out the common factors,
then multiply:
6 14 2 3 2 7 4
7 15
7 3 5
5
When we multiply rational numbers, we use the following definition.
Multiplication of Rational Numbers
a
b
If and c are rational numbers, then a c ac.
d
b d
bd
We multiply rational expressions in the same way that we multiply rational
numbers: Factor all polynomials, divide out the common factors, then multiply the
remaining factors.