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Elementary Algebra 2e

8.2 Multiply and Divide Rational Expressions

Elementary Algebra 2e8.2 Multiply and Divide Rational Expressions
  1. Preface
  2. 1 Foundations
    1. Introduction
    2. 1.1 Introduction to Whole Numbers
    3. 1.2 Use the Language of Algebra
    4. 1.3 Add and Subtract Integers
    5. 1.4 Multiply and Divide Integers
    6. 1.5 Visualize Fractions
    7. 1.6 Add and Subtract Fractions
    8. 1.7 Decimals
    9. 1.8 The Real Numbers
    10. 1.9 Properties of Real Numbers
    11. 1.10 Systems of Measurement
    12. Key Terms
    13. Key Concepts
    14. Exercises
      1. Review Exercises
      2. Practice Test
  3. 2 Solving Linear Equations and Inequalities
    1. Introduction
    2. 2.1 Solve Equations Using the Subtraction and Addition Properties of Equality
    3. 2.2 Solve Equations using the Division and Multiplication Properties of Equality
    4. 2.3 Solve Equations with Variables and Constants on Both Sides
    5. 2.4 Use a General Strategy to Solve Linear Equations
    6. 2.5 Solve Equations with Fractions or Decimals
    7. 2.6 Solve a Formula for a Specific Variable
    8. 2.7 Solve Linear Inequalities
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  4. 3 Math Models
    1. Introduction
    2. 3.1 Use a Problem-Solving Strategy
    3. 3.2 Solve Percent Applications
    4. 3.3 Solve Mixture Applications
    5. 3.4 Solve Geometry Applications: Triangles, Rectangles, and the Pythagorean Theorem
    6. 3.5 Solve Uniform Motion Applications
    7. 3.6 Solve Applications with Linear Inequalities
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  5. 4 Graphs
    1. Introduction
    2. 4.1 Use the Rectangular Coordinate System
    3. 4.2 Graph Linear Equations in Two Variables
    4. 4.3 Graph with Intercepts
    5. 4.4 Understand Slope of a Line
    6. 4.5 Use the Slope-Intercept Form of an Equation of a Line
    7. 4.6 Find the Equation of a Line
    8. 4.7 Graphs of Linear Inequalities
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  6. 5 Systems of Linear Equations
    1. Introduction
    2. 5.1 Solve Systems of Equations by Graphing
    3. 5.2 Solving Systems of Equations by Substitution
    4. 5.3 Solve Systems of Equations by Elimination
    5. 5.4 Solve Applications with Systems of Equations
    6. 5.5 Solve Mixture Applications with Systems of Equations
    7. 5.6 Graphing Systems of Linear Inequalities
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  7. 6 Polynomials
    1. Introduction
    2. 6.1 Add and Subtract Polynomials
    3. 6.2 Use Multiplication Properties of Exponents
    4. 6.3 Multiply Polynomials
    5. 6.4 Special Products
    6. 6.5 Divide Monomials
    7. 6.6 Divide Polynomials
    8. 6.7 Integer Exponents and Scientific Notation
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  8. 7 Factoring
    1. Introduction
    2. 7.1 Greatest Common Factor and Factor by Grouping
    3. 7.2 Factor Trinomials of the Form x2+bx+c
    4. 7.3 Factor Trinomials of the Form ax2+bx+c
    5. 7.4 Factor Special Products
    6. 7.5 General Strategy for Factoring Polynomials
    7. 7.6 Quadratic Equations
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  9. 8 Rational Expressions and Equations
    1. Introduction
    2. 8.1 Simplify Rational Expressions
    3. 8.2 Multiply and Divide Rational Expressions
    4. 8.3 Add and Subtract Rational Expressions with a Common Denominator
    5. 8.4 Add and Subtract Rational Expressions with Unlike Denominators
    6. 8.5 Simplify Complex Rational Expressions
    7. 8.6 Solve Rational Equations
    8. 8.7 Solve Proportion and Similar Figure Applications
    9. 8.8 Solve Uniform Motion and Work Applications
    10. 8.9 Use Direct and Inverse Variation
    11. Key Terms
    12. Key Concepts
    13. Exercises
      1. Review Exercises
      2. Practice Test
  10. 9 Roots and Radicals
    1. Introduction
    2. 9.1 Simplify and Use Square Roots
    3. 9.2 Simplify Square Roots
    4. 9.3 Add and Subtract Square Roots
    5. 9.4 Multiply Square Roots
    6. 9.5 Divide Square Roots
    7. 9.6 Solve Equations with Square Roots
    8. 9.7 Higher Roots
    9. 9.8 Rational Exponents
    10. Key Terms
    11. Key Concepts
    12. Exercises
      1. Review Exercises
      2. Practice Test
  11. 10 Quadratic Equations
    1. Introduction
    2. 10.1 Solve Quadratic Equations Using the Square Root Property
    3. 10.2 Solve Quadratic Equations by Completing the Square
    4. 10.3 Solve Quadratic Equations Using the Quadratic Formula
    5. 10.4 Solve Applications Modeled by Quadratic Equations
    6. 10.5 Graphing Quadratic Equations in Two Variables
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  12. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
    10. Chapter 10
  13. Index

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: 2x298.2x298.
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: 10p225pq15q2.10p225pq15q2.
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 q0ands0q0ands0, 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, x0x0 and y0y0.

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: 2xx2-7x12·x296x2.2xx2-7x12·x296x2.

Try It 8.37

Mulitply: 5xx2+5x+6·x2410x.5xx2+5x+6·x2410x.

Try It 8.38

Mulitply: 9x2x2+11x+30·x2363x2.9x2x2+11x+30·x2363x2.

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: n27nn2+2n+1·n+12n.n27nn2+2n+1·n+12n.

Try It 8.39

Multiply: x225x23x10·x+2x.x225x23x10·x+2x.

Try It 8.40

Multiply: x24xx2+5x+6·x+2x.x24xx2+5x+6·x+2x.

Example 8.21

Multiply: 164x2x12·x25x6x216.164x2x12·x25x6x216.

Try It 8.41

Multiply: 12x6x2x2+8x·x2+11x+24x24.12x6x2x2+8x·x2+11x+24x24.

Try It 8.42

Multiply: 9v3v29v+36·v2+7v+12v29.9v3v29v+36·v2+7v+12v29.

Example 8.22

Multiply: 2x6x28x+15·x2252x+10.2x6x28x+15·x2252x+10.

Try It 8.43

Multiply: 3a21a29a+14·a243a+6.3a21a29a+14·a243a+6.

Try It 8.44

Multiply: b2bb2+9b10·b2100b210b.b2bb2+9b10·b2100b210b.

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 q0,r0,s0q0,r0,s0, 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+96x÷x281x6.x+96x÷x281x6.

Try It 8.45

Divide: c+35c÷c29c5.c+35c÷c29c5.

Try It 8.46

Divide: 2dd4÷4d24d.2dd4÷4d24d.

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: 3n2n24n÷9n245nn27n+10.3n2n24n÷9n245nn27n+10.

Try It 8.47

Divide: 2m2m28m÷8m2+24mm2+m6.2m2m28m÷8m2+24mm2+m6.

Try It 8.48

Divide: 15n23n2+33n÷5n5n2+9n22.15n23n2+33n÷5n5n2+9n22.

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+5x12x216÷2x213x+15x28x+16.2x2+5x12x216÷2x213x+15x28x+16.

Try It 8.49

Divide: 3a28a3a225÷3a214a5a2+10a+25.3a28a3a225÷3a214a5a2+10a+25.

Try It 8.50

Divide: 4b2+7b21b2÷4b2+15b4b22b+1.4b2+7b21b2÷4b2+15b4b22b+1.

Example 8.26

Divide: p3+q32p2+2pq+2q2÷p2q26.p3+q32p2+2pq+2q2÷p2q26.

Try It 8.51

Divide: x383x26x+12÷x246.x383x26x+12÷x246.

Try It 8.52

Divide: 2z2z21÷z3z2+zz31.2z2z21÷z3z2+zz31.

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: a2b23ab÷(a2+2ab+b2).a2b23ab÷(a2+2ab+b2).

Try It 8.53

Divide: 2x214x164÷(x2+2x+1).2x214x164÷(x2+2x+1).

Try It 8.54

Divide: y26y+8y24y÷(3y212y).y26y+8y24y÷(3y212y).

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

Example 8.28

Divide: 6x27x+24x82x27x+3x25x+6.6x27x+24x82x27x+3x25x+6.

Try It 8.55

Divide: 3x2+7x+24x+243x214x5x2+x30.3x2+7x+24x+243x214x5x2+x30.

Try It 8.56

Divide: y2362y2+11y62y22y608y4.y2362y2+11y62y22y608y4.

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: 3x64x4·x2+2x3x23x10÷2x+128x+16.3x64x4·x2+2x3x23x10÷2x+128x+16.

Try It 8.57

Divide: 4m+43m15·m23m10m24m32÷12m366m48.4m+43m15·m23m10m24m32÷12m366m48.

Try It 8.58

Divide: 2n2+10nn1÷n2+10n+24n2+8n9·n+48n2+12n.2n2+10nn1÷n2+10n+24n2+8n9·n+48n2+12n.

Section 8.2 Exercises

Practice Makes Perfect

Multiply Rational Expressions

In the following exercises, multiply.

73.

1216·4101216·410

74.

325·1624325·1624

75.

1810·4301810·430

76.

2136·45242136·4524

77.

5x2y412xy3·6x220y25x2y412xy3·6x220y2

78.

8w3y9y2·3y4w48w3y9y2·3y4w4

79.

12a3bb2·2ab29b312a3bb2·2ab29b3

80.

4mn25n3·mn38m2n24mn25n3·mn38m2n2

81.

5p2p25p36·p21610p5p2p25p36·p21610p

82.

3q2q2+q6·q299q3q2q2+q6·q299q

83.

4rr23r10·r2258r24rr23r10·r2258r2

84.

ss29s+14·s2497s2ss29s+14·s2497s2

85.

x27xx2+6x+9·x+34xx27xx2+6x+9·x+34x

86.

2y210yy2+10y+25·y+56y2y210yy2+10y+25·y+56y

87.

z2+3zz23z4·z4z2z2+3zz23z4·z4z2

88.

2a2+8aa29a+20·a5a22a2+8aa29a+20·a5a2

89.

284b3b3·b2+8b9b249284b3b3·b2+8b9b249

90.

18c2c26c+30·c2+7c+10c28118c2c26c+30·c2+7c+10c281

91.

35d7d2d2+7d·d2+12d+35d22535d7d2d2+7d·d2+12d+35d225

92.

72m12m28m+32·m2+10m+24m23672m12m28m+32·m2+10m+24m236

93.

4n+20n2+n20·n2164n+164n+20n2+n20·n2164n+16

94.

6p26pp2+7p18·p2813p227p6p26pp2+7p18·p2813p227p

95.

q22qq2+6q16·q264q28qq22qq2+6q16·q264q28q

96.

2r22rr2+4r5·r2252r210r2r22rr2+4r5·r2252r210r

Divide Rational Expressions

In the following exercises, divide.

97.

t63t÷t5t29t63t÷t5t29

98.

v511v÷v225v11v511v÷v225v11

99.

10+ww8÷100w28w10+ww8÷100w28w

100.

7+xx6÷49xx+627+xx6÷49xx+62

101.

27y23y21÷3y2+18y2+13y+4227y23y21÷3y2+18y2+13y+42

102.

24z22z8÷4z28z211z+2824z22z8÷4z28z211z+28

103.

16a24a+36÷4a224aa2+4a4516a24a+36÷4a224aa2+4a45

104.

24b22b4÷12b2+36bb211b+1824b22b4÷12b2+36bb211b+18

105.

5c2+9c2c24÷5c216c+3c2+4c+45c2+9c2c24÷5c216c+3c2+4c+4

106.

2d2+d3d216÷2d29d18d28d+162d2+d3d216÷2d29d18d28d+16

107.

6m211m29m2÷6m2+25m+4m26m+96m211m29m2÷6m2+25m+4m26m+9

108.

2n23n1425n2÷2n213n+21n210n+252n23n1425n2÷2n213n+21n210n+25

109.

3s2s216÷s3+4s2+16ss3643s2s216÷s3+4s2+16ss364

110.

r2915÷r3275r2+15r+45r2915÷r3275r2+15r+45

111.

p3+q33p2+3pq+3q2÷p2q212p3+q33p2+3pq+3q2÷p2q212

112.

v38w32v2+4vw+8w2÷v24w24v38w32v2+4vw+8w2÷v24w24

113.

t292t÷(t26t+9)t292t÷(t26t+9)

114.

x2+3x104x÷(2x2+20x+50)x2+3x104x÷(2x2+20x+50)

115.

2y210yz48z22y1÷(4y232yz)2y210yz48z22y1÷(4y232yz)

116.

2m298n22m+6÷(m27mn)2m298n22m+6÷(m27mn)

117.

2a2a215a+20a2+7a+12a2+8a+162a2a215a+20a2+7a+12a2+8a+16

118.

3b2+2b812b+183b2+2b82b27b153b2+2b812b+183b2+2b82b27b15

119.

12c2122c23c+14c+46c213c+512c2122c23c+14c+46c213c+5

120.

4d2+7d235d+10d247d212d44d2+7d235d+10d247d212d4

121.

10m2+80m3m9·m2+4m21m29m+2010m2+80m3m9·m2+4m21m29m+20
÷5m2+10m2m10÷5m2+10m2m10

122.

4n2+32n3n+2·3n2n2n2+n304n2+32n3n+2·3n2n2n2+n30
÷108n224nn+6÷108n224nn+6

123.

12p2+3pp+3÷p2+2p63p2p1212p2+3pp+3÷p2+2p63p2p12
·p79p39p2·p79p39p2

124.

6q+39q29q÷q2+14q+33q2+4q56q+39q29q÷q2+14q+33q2+4q5
·4q2+12q12q+6·4q2+12q12q+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·w1w+m1.ww+m·w1w+m1.

  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 nn3·9n+3nn3·9n+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 x21x÷(x+1)x21x÷(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.

The above image is a table with four columns and four rows. The first row is the header row. The first header is labeled “I can…”, the second “Confidently”, the third, “With some help”, and the fourth “No – I don’t get it!”. In the first column under “I can”, the next row reads multiply rational expressions.”, the next row reads “divide rational expressions.”, the last row reads “after reviewing this checklist, what will you do to become confident for all objectives?” The remaining columns are blank.

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

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