Elementary Algebra 2e

# 8.2Multiply and Divide Rational Expressions

Elementary Algebra 2e8.2 Multiply and Divide Rational Expressions

### Learning Objectives

By the end of this section, you will be able to:

• Multiply rational expressions
• Divide rational expressions

### Be Prepared 8.4

Before you get started, take this readiness quiz.

If you miss a problem, go back to the section listed and review the material.

Multiply: $1415·635.1415·635.$
If you missed this problem, review Example 1.68.

### Be Prepared 8.5

Divide: $1415÷635.1415÷635.$
If you missed this problem, review Example 1.71.

### Be Prepared 8.6

Factor completely: $2x2−98.2x2−98.$
If you missed this problem, review Example 7.62.

### Be Prepared 8.7

Factor completely: $10n3+10.10n3+10.$
If you missed this problem, review Example 7.65.

### Be Prepared 8.8

Factor completely: $10p2−25pq−15q2.10p2−25pq−15q2.$
If you missed this problem, review Example 7.68.

### Multiply Rational Expressions

To multiply rational expressions, we do just what we did with numerical fractions. We multiply the numerators and multiply the denominators. Then, if there are any common factors, we remove them to simplify the result.

### Multiplication of Rational Expressions

If $p,q,r,sp,q,r,s$ are polynomials where $q≠0ands≠0q≠0ands≠0$, then

$pq·rs=prqspq·rs=prqs$

To multiply rational expressions, multiply the numerators and multiply the denominators.

We’ll do the first example with numerical fractions to remind us of how we multiplied fractions without variables.

### Example 8.17

Multiply: $1028·815.1028·815.$

### Try It 8.33

Mulitply: $610·1512.610·1512.$

### Try It 8.34

Mulitply: $2015·68.2015·68.$

Remember, throughout this chapter, we will assume that all numerical values that would make the denominator be zero are excluded. We will not write the restrictions for each rational expression, but keep in mind that the denominator can never be zero. So in this next example, $x≠0x≠0$ and $y≠0y≠0$.

### Example 8.18

Mulitply: $2x3y2·6xy3x2y.2x3y2·6xy3x2y.$

### Try It 8.35

Mulitply: $3pqq2·5p2q6pq.3pqq2·5p2q6pq.$

### Try It 8.36

Mulitply: $6x3y7x2·2xy3x2y.6x3y7x2·2xy3x2y.$

### Example 8.19

#### How to Multiply Rational Expressions

Mulitply: $2xx2-7x+12·x2−96x2.2xx2-7x+12·x2−96x2.$

### Try It 8.37

Mulitply: $5xx2+5x+6·x2−410x.5xx2+5x+6·x2−410x.$

### Try It 8.38

Mulitply: $9x2x2+11x+30·x2−363x2.9x2x2+11x+30·x2−363x2.$

### How To

#### Multiply a rational expression.

1. Step 1. Factor each numerator and denominator completely.
2. Step 2. Multiply the numerators and denominators.
3. Step 3. Simplify by dividing out common factors.

### Example 8.20

Multiply: $n2−7nn2+2n+1·n+12n.n2−7nn2+2n+1·n+12n.$

### Try It 8.39

Multiply: $x2−25x2−3x−10·x+2x.x2−25x2−3x−10·x+2x.$

### Try It 8.40

Multiply: $x2−4xx2+5x+6·x+2x.x2−4xx2+5x+6·x+2x.$

### Example 8.21

Multiply: $16−4x2x−12·x2−5x−6x2−16.16−4x2x−12·x2−5x−6x2−16.$

### Try It 8.41

Multiply: $12x−6x2x2+8x·x2+11x+24x2−4.12x−6x2x2+8x·x2+11x+24x2−4.$

### Try It 8.42

Multiply: $9v−3v29v+36·v2+7v+12v2−9.9v−3v29v+36·v2+7v+12v2−9.$

### Example 8.22

Multiply: $2x−6x2−8x+15·x2−252x+10.2x−6x2−8x+15·x2−252x+10.$

### Try It 8.43

Multiply: $3a−21a2−9a+14·a2−43a+6.3a−21a2−9a+14·a2−43a+6.$

### Try It 8.44

Multiply: $b2−bb2+9b−10·b2−100b2−10b.b2−bb2+9b−10·b2−100b2−10b.$

### Divide Rational Expressions

To divide rational expressions we multiply the first fraction by the reciprocal of the second, just like we did for numerical fractions.

Remember, the reciprocal of $abab$ is $baba$. To find the reciprocal we simply put the numerator in the denominator and the denominator in the numerator. We “flip” the fraction.

### Division of Rational Expressions

If $p,q,r,sp,q,r,s$ are polynomials where $q≠0,r≠0,s≠0q≠0,r≠0,s≠0$, then

$pq÷rs=pq·srpq÷rs=pq·sr$

To divide rational expressions multiply the first fraction by the reciprocal of the second.

### Example 8.23

#### How to Divide Rational Expressions

Divide: $x+96−x÷x2−81x−6.x+96−x÷x2−81x−6.$

### Try It 8.45

Divide: $c+35−c÷c2−9c−5.c+35−c÷c2−9c−5.$

### Try It 8.46

Divide: $2−dd−4÷4−d24−d.2−dd−4÷4−d24−d.$

### How To

#### Divide rational expressions.

1. Step 1. Rewrite the division as the product of the first rational expression and the reciprocal of the second.
2. Step 2. Factor the numerators and denominators completely.
3. Step 3. Multiply the numerators and denominators together.
4. Step 4. Simplify by dividing out common factors.

### Example 8.24

Divide: $3n2n2−4n÷9n2−45nn2−7n+10.3n2n2−4n÷9n2−45nn2−7n+10.$

### Try It 8.47

Divide: $2m2m2−8m÷8m2+24mm2+m−6.2m2m2−8m÷8m2+24mm2+m−6.$

### Try It 8.48

Divide: $15n23n2+33n÷5n−5n2+9n−22.15n23n2+33n÷5n−5n2+9n−22.$

Remember, first rewrite the division as multiplication of the first expression by the reciprocal of the second. Then factor everything and look for common factors.

### Example 8.25

Divide: $2x2+5x−12x2−16÷2x2−13x+15x2−8x+16.2x2+5x−12x2−16÷2x2−13x+15x2−8x+16.$

### Try It 8.49

Divide: $3a2−8a−3a2−25÷3a2−14a−5a2+10a+25.3a2−8a−3a2−25÷3a2−14a−5a2+10a+25.$

### Try It 8.50

Divide: $4b2+7b−21−b2÷4b2+15b−4b2−2b+1.4b2+7b−21−b2÷4b2+15b−4b2−2b+1.$

### Example 8.26

Divide: $p3+q32p2+2pq+2q2÷p2−q26.p3+q32p2+2pq+2q2÷p2−q26.$

### Try It 8.51

Divide: $x3−83x2−6x+12÷x2−46.x3−83x2−6x+12÷x2−46.$

### Try It 8.52

Divide: $2z2z2−1÷z3−z2+zz3−1.2z2z2−1÷z3−z2+zz3−1.$

Before doing the next example, let’s look at how we divide a fraction by a whole number. When we divide $35÷435÷4$, we first write 4 as a fraction so that we can find its reciprocal.

$35÷435÷4135·1435÷435÷4135·14$

We do the same thing when we divide rational expressions.

### Example 8.27

Divide: $a2−b23ab÷(a2+2ab+b2).a2−b23ab÷(a2+2ab+b2).$

### Try It 8.53

Divide: $2x2−14x−164÷(x2+2x+1).2x2−14x−164÷(x2+2x+1).$

### Try It 8.54

Divide: $y2−6y+8y2−4y÷(3y2−12y).y2−6y+8y2−4y÷(3y2−12y).$

Remember a fraction bar means division. A complex fraction is another way of writing division of two fractions.

### Example 8.28

Divide: $6x2−7x+24x−82x2−7x+3x2−5x+6.6x2−7x+24x−82x2−7x+3x2−5x+6.$

### Try It 8.55

Divide: $3x2+7x+24x+243x2−14x−5x2+x−30.3x2+7x+24x+243x2−14x−5x2+x−30.$

### Try It 8.56

Divide: $y2−362y2+11y−62y2−2y−608y−4.y2−362y2+11y−62y2−2y−608y−4.$

If we have more than two rational expressions to work with, we still follow the same procedure. The first step will be to rewrite any division as multiplication by the reciprocal. Then we factor and multiply.

### Example 8.29

Divide: $3x−64x−4·x2+2x−3x2−3x−10÷2x+128x+16.3x−64x−4·x2+2x−3x2−3x−10÷2x+128x+16.$

### Try It 8.57

Divide: $4m+43m−15·m2−3m−10m2−4m−32÷12m−366m−48.4m+43m−15·m2−3m−10m2−4m−32÷12m−366m−48.$

### Try It 8.58

Divide: $2n2+10nn−1÷n2+10n+24n2+8n−9·n+48n2+12n.2n2+10nn−1÷n2+10n+24n2+8n−9·n+48n2+12n.$

### Section 8.2 Exercises

#### Practice Makes Perfect

Multiply Rational Expressions

In the following exercises, multiply.

73.

$12 16 · 4 10 12 16 · 4 10$

74.

$32 5 · 16 24 32 5 · 16 24$

75.

$18 10 · 4 30 18 10 · 4 30$

76.

$21 36 · 45 24 21 36 · 45 24$

77.

$5 x 2 y 4 12 x y 3 · 6 x 2 20 y 2 5 x 2 y 4 12 x y 3 · 6 x 2 20 y 2$

78.

$8 w 3 y 9 y 2 · 3 y 4 w 4 8 w 3 y 9 y 2 · 3 y 4 w 4$

79.

$12 a 3 b b 2 · 2 a b 2 9 b 3 12 a 3 b b 2 · 2 a b 2 9 b 3$

80.

$4 m n 2 5 n 3 · m n 3 8 m 2 n 2 4 m n 2 5 n 3 · m n 3 8 m 2 n 2$

81.

$5 p 2 p 2 − 5 p − 36 · p 2 − 16 10 p 5 p 2 p 2 − 5 p − 36 · p 2 − 16 10 p$

82.

$3 q 2 q 2 + q − 6 · q 2 − 9 9 q 3 q 2 q 2 + q − 6 · q 2 − 9 9 q$

83.

$4 r r 2 − 3 r − 10 · r 2 − 25 8 r 2 4 r r 2 − 3 r − 10 · r 2 − 25 8 r 2$

84.

$s s 2 − 9 s + 14 · s 2 − 49 7 s 2 s s 2 − 9 s + 14 · s 2 − 49 7 s 2$

85.

$x 2 − 7 x x 2 + 6 x + 9 · x + 3 4 x x 2 − 7 x x 2 + 6 x + 9 · x + 3 4 x$

86.

$2 y 2 − 10 y y 2 + 10 y + 25 · y + 5 6 y 2 y 2 − 10 y y 2 + 10 y + 25 · y + 5 6 y$

87.

$z 2 + 3 z z 2 − 3 z − 4 · z − 4 z 2 z 2 + 3 z z 2 − 3 z − 4 · z − 4 z 2$

88.

$2 a 2 + 8 a a 2 − 9 a + 20 · a − 5 a 2 2 a 2 + 8 a a 2 − 9 a + 20 · a − 5 a 2$

89.

$28 − 4 b 3 b − 3 · b 2 + 8 b − 9 b 2 − 49 28 − 4 b 3 b − 3 · b 2 + 8 b − 9 b 2 − 49$

90.

$18 c − 2 c 2 6 c + 30 · c 2 + 7 c + 10 c 2 − 81 18 c − 2 c 2 6 c + 30 · c 2 + 7 c + 10 c 2 − 81$

91.

$35 d − 7 d 2 d 2 + 7 d · d 2 + 12 d + 35 d 2 − 25 35 d − 7 d 2 d 2 + 7 d · d 2 + 12 d + 35 d 2 − 25$

92.

$72 m − 12 m 2 8 m + 32 · m 2 + 10 m + 24 m 2 − 36 72 m − 12 m 2 8 m + 32 · m 2 + 10 m + 24 m 2 − 36$

93.

$4 n + 20 n 2 + n − 20 · n 2 − 16 4 n + 16 4 n + 20 n 2 + n − 20 · n 2 − 16 4 n + 16$

94.

$6 p 2 − 6 p p 2 + 7 p − 18 · p 2 − 81 3 p 2 − 27 p 6 p 2 − 6 p p 2 + 7 p − 18 · p 2 − 81 3 p 2 − 27 p$

95.

$q 2 − 2 q q 2 + 6 q − 16 · q 2 − 64 q 2 − 8 q q 2 − 2 q q 2 + 6 q − 16 · q 2 − 64 q 2 − 8 q$

96.

$2 r 2 − 2 r r 2 + 4 r − 5 · r 2 − 25 2 r 2 − 10 r 2 r 2 − 2 r r 2 + 4 r − 5 · r 2 − 25 2 r 2 − 10 r$

Divide Rational Expressions

In the following exercises, divide.

97.

$t − 6 3 − t ÷ t − 5 t 2 − 9 t − 6 3 − t ÷ t − 5 t 2 − 9$

98.

$v − 5 11 − v ÷ v 2 − 25 v − 11 v − 5 11 − v ÷ v 2 − 25 v − 11$

99.

$10 + w w − 8 ÷ 100 − w 2 8 − w 10 + w w − 8 ÷ 100 − w 2 8 − w$

100.

$7 + x x − 6 ÷ 49 − x x + 6 2 7 + x x − 6 ÷ 49 − x x + 6 2$

101.

$27 y 2 3 y − 21 ÷ 3 y 2 + 18 y 2 + 13 y + 42 27 y 2 3 y − 21 ÷ 3 y 2 + 18 y 2 + 13 y + 42$

102.

$24 z 2 2 z − 8 ÷ 4 z − 28 z 2 − 11 z + 28 24 z 2 2 z − 8 ÷ 4 z − 28 z 2 − 11 z + 28$

103.

$16 a 2 4 a + 36 ÷ 4 a 2 − 24 a a 2 + 4 a − 45 16 a 2 4 a + 36 ÷ 4 a 2 − 24 a a 2 + 4 a − 45$

104.

$24 b 2 2 b − 4 ÷ 12 b 2 + 36 b b 2 − 11 b + 18 24 b 2 2 b − 4 ÷ 12 b 2 + 36 b b 2 − 11 b + 18$

105.

$5 c 2 + 9 c − 2 c 2 − 4 ÷ 5 c 2 − 16 c + 3 c 2 + 4 c + 4 5 c 2 + 9 c − 2 c 2 − 4 ÷ 5 c 2 − 16 c + 3 c 2 + 4 c + 4$

106.

$2 d 2 + d − 3 d 2 − 16 ÷ 2 d 2 − 9 d − 18 d 2 − 8 d + 16 2 d 2 + d − 3 d 2 − 16 ÷ 2 d 2 − 9 d − 18 d 2 − 8 d + 16$

107.

$6 m 2 − 11 m − 2 9 − m 2 ÷ 6 m 2 + 25 m + 4 m 2 − 6 m + 9 6 m 2 − 11 m − 2 9 − m 2 ÷ 6 m 2 + 25 m + 4 m 2 − 6 m + 9$

108.

$2 n 2 − 3 n − 14 25 − n 2 ÷ 2 n 2 − 13 n + 21 n 2 − 10 n + 25 2 n 2 − 3 n − 14 25 − n 2 ÷ 2 n 2 − 13 n + 21 n 2 − 10 n + 25$

109.

$3 s 2 s 2 − 16 ÷ s 3 + 4 s 2 + 16 s s 3 − 64 3 s 2 s 2 − 16 ÷ s 3 + 4 s 2 + 16 s s 3 − 64$

110.

$r 2 − 9 15 ÷ r 3 − 27 5 r 2 + 15 r + 45 r 2 − 9 15 ÷ r 3 − 27 5 r 2 + 15 r + 45$

111.

$p 3 + q 3 3 p 2 + 3 p q + 3 q 2 ÷ p 2 − q 2 12 p 3 + q 3 3 p 2 + 3 p q + 3 q 2 ÷ p 2 − q 2 12$

112.

$v 3 − 8 w 3 2 v 2 + 4 v w + 8 w 2 ÷ v 2 − 4 w 2 4 v 3 − 8 w 3 2 v 2 + 4 v w + 8 w 2 ÷ v 2 − 4 w 2 4$

113.

$t 2 − 9 2 t ÷ ( t 2 − 6 t + 9 ) t 2 − 9 2 t ÷ ( t 2 − 6 t + 9 )$

114.

$x 2 + 3 x − 10 4 x ÷ ( 2 x 2 + 20 x + 50 ) x 2 + 3 x − 10 4 x ÷ ( 2 x 2 + 20 x + 50 )$

115.

$2 y 2 − 10 y z − 48 z 2 2 y − 1 ÷ ( 4 y 2 − 32 y z ) 2 y 2 − 10 y z − 48 z 2 2 y − 1 ÷ ( 4 y 2 − 32 y z )$

116.

$2 m 2 − 98 n 2 2 m + 6 ÷ ( m 2 − 7 m n ) 2 m 2 − 98 n 2 2 m + 6 ÷ ( m 2 − 7 m n )$

117.

$2 a 2 − a − 21 5 a + 20 a 2 + 7 a + 12 a 2 + 8 a + 16 2 a 2 − a − 21 5 a + 20 a 2 + 7 a + 12 a 2 + 8 a + 16$

118.

$3 b 2 + 2 b − 8 12 b + 18 3 b 2 + 2 b − 8 2 b 2 − 7 b − 15 3 b 2 + 2 b − 8 12 b + 18 3 b 2 + 2 b − 8 2 b 2 − 7 b − 15$

119.

$12 c 2 − 12 2 c 2 − 3 c + 1 4 c + 4 6 c 2 − 13 c + 5 12 c 2 − 12 2 c 2 − 3 c + 1 4 c + 4 6 c 2 − 13 c + 5$

120.

$4 d 2 + 7 d − 2 35 d + 10 d 2 − 4 7 d 2 − 12 d − 4 4 d 2 + 7 d − 2 35 d + 10 d 2 − 4 7 d 2 − 12 d − 4$

121.

$10 m 2 + 80 m 3 m − 9 · m 2 + 4 m − 21 m 2 − 9 m + 20 10 m 2 + 80 m 3 m − 9 · m 2 + 4 m − 21 m 2 − 9 m + 20$
$÷ 5 m 2 + 10 m 2 m − 10 ÷ 5 m 2 + 10 m 2 m − 10$

122.

$4 n 2 + 32 n 3 n + 2 · 3 n 2 − n − 2 n 2 + n − 30 4 n 2 + 32 n 3 n + 2 · 3 n 2 − n − 2 n 2 + n − 30$
$÷ 108 n 2 − 24 n n + 6 ÷ 108 n 2 − 24 n n + 6$

123.

$12 p 2 + 3 p p + 3 ÷ p 2 + 2 p − 63 p 2 − p − 12 12 p 2 + 3 p p + 3 ÷ p 2 + 2 p − 63 p 2 − p − 12$
$· p − 7 9 p 3 − 9 p 2 · p − 7 9 p 3 − 9 p 2$

124.

$6 q + 3 9 q 2 − 9 q ÷ q 2 + 14 q + 33 q 2 + 4 q − 5 6 q + 3 9 q 2 − 9 q ÷ q 2 + 14 q + 33 q 2 + 4 q − 5$
$· 4 q 2 + 12 q 12 q + 6 · 4 q 2 + 12 q 12 q + 6$

#### Everyday Math

125.

Probability The director of large company is interviewing applicants for two identical jobs. If $w=w=$ the number of women applicants and $m=m=$ the number of men applicants, then the probability that two women are selected for the jobs is $ww+m·w−1w+m−1.ww+m·w−1w+m−1.$

1. Simplify the probability by multiplying the two rational expressions.
2. Find the probability that two women are selected when $w=5w=5$ and $m=10m=10$.
126.

Area of a triangle The area of a triangle with base b and height h is $bh2.bh2.$ If the triangle is stretched to make a new triangle with base and height three times as much as in the original triangle, the area is $9bh2.9bh2.$ Calculate how the area of the new triangle compares to the area of the original triangle by dividing $9bh29bh2$ by $bh2bh2$.

#### Writing Exercises

127.
1. Multiply $74·91074·910$ and explain all your steps.
2. Multiply $nn−3·9n+3nn−3·9n+3$ and explain all your steps.
3. Evaluate your answer to part (b) when $n=7.n=7.$ Did you get the same answer you got in part (a)? Why or why not?
128.
1. Divide $245÷6245÷6$ and explain all your steps.
2. Divide $x2−1x÷(x+1)x2−1x÷(x+1)$ and explain all your steps.
3. Evaluate your answer to part (b) when $x=5.x=5.$ Did you get the same answer you got in part (a)? Why or why not?

#### Self Check

After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

After reviewing this checklist, what will you do to become confident for all objectives?

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