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Divide: 6 x 2 7 x + 2 4 x 8 2 x 2 7 x + 3 x 2 5 x + 6 .

Solution

6 x 2 7 x + 2 4 x 8 2 x 2 7 x + 3 x 2 5 x + 6 Rewrite with a division sign. 6 x 2 7 x + 2 4 x 8 ÷ 2 x 2 7 x + 3 x 2 5 x + 6 Rewrite as product of first times reciprocal of second. 6 x 2 7 x + 2 4 x 8 · x 2 5 x + 6 2 x 2 7 x + 3 Factor the numerators and the denominators, and then multiply. ( 2 x 1 ) ( 3 x 2 ) ( x 2 ) ( x 3 ) 4 ( x 2 ) ( 2 x 1 ) ( x 3 ) Simplify by dividing out common factors. ( 2 x 1 ) ( 3 x 2 ) ( x 2 ) ( x 3 ) 4 ( x 2 ) ( 2 x 1 ) ( x 3 ) Simplify. 3 x 2 4

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Divide: 3 x 2 + 7 x + 2 4 x + 24 3 x 2 14 x 5 x 2 + x 30 .

x + 2 4

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Divide: y 2 36 2 y 2 + 11 y 6 2 y 2 2 y 60 8 y 4 .

2 y + 5

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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.

Divide: 3 x 6 4 x 4 · x 2 + 2 x 3 x 2 3 x 10 ÷ 2 x + 12 8 x + 16 .

Solution

.
Rewrite the division as multiplication by the reciprocal. .
Factor the numerators and the denominators, and then multiply. .
Simplify by dividing out common factors. .
Simplify. .

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Divide: 4 m + 4 3 m 15 · m 2 3 m 10 m 2 4 m 32 ÷ 12 m 36 6 m 48 .

2 ( m + 1 ) ( m + 2 ) 3 ( m + 4 ) ( m 3 )

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Divide: 2 n 2 + 10 n n 1 ÷ n 2 + 10 n + 24 n 2 + 8 n 9 · n + 4 8 n 2 + 12 n .

( n + 5 ) ( n + 9 ) 2 ( n + 6 ) ( 2 n + 3 )

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Key concepts

  • Multiplication of Rational Expressions
    • If p , q , r , s are polynomials where q 0 , s 0 , then p q · r s = p r q s .
    • To multiply rational expressions, multiply the numerators and multiply the denominators
  • Multiply a Rational Expression
    1. Factor each numerator and denominator completely.
    2. Multiply the numerators and denominators.
    3. Simplify by dividing out common factors.
  • Division of Rational Expressions
    • If p , q , r , s are polynomials where q 0 , r 0 , s 0 , then p q ÷ r s = p q · s r .
    • To divide rational expressions multiply the first fraction by the reciprocal of the second.
  • Divide Rational Expressions
    1. Rewrite the division as the product of the first rational expression and the reciprocal of the second.
    2. Factor the numerators and denominators completely.
    3. Multiply the numerators and denominators together.

    4. Simplify by dividing out common factors.

Practice makes perfect

Multiply Rational Expressions

In the following exercises, multiply.

5 x 2 y 4 12 x y 3 · 6 x 2 20 y 2

x 3 8 y

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8 w 3 y 9 y 2 · 3 y 4 w 4

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12 a 3 b b 2 · 2 a b 2 9 b 3

8 a b 3

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4 m n 2 5 n 3 · m n 3 8 m 2 n 2

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5 p 2 p 2 5 p 36 · p 2 16 10 p

p ( p 4 ) 2 ( p 9 )

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3 q 2 q 2 + q 6 · q 2 9 9 q

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4 r r 2 3 r 10 · r 2 25 8 r 2

r + 5 2 r ( r + 2 )

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s s 2 9 s + 14 · s 2 49 7 s 2

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x 2 7 x x 2 + 6 x + 9 · x + 3 4 x

x 7 4 ( x + 3 )

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

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z 2 + 3 z z 2 3 z 4 · z 4 z 2

z + 3 z ( z + 1 )

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2 a 2 + 8 a a 2 9 a + 20 · a 5 a 2

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28 4 b 3 b 3 · b 2 + 8 b 9 b 2 49

4 ( b + 9 ) 3 ( b + 7 )

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

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

−7

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

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4 n + 20 n 2 + n 20 · n 2 16 4 n + 16

1

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

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q 2 2 q q 2 + 6 q 16 · q 2 64 q 2 8 q

1

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

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Divide Rational Expressions

In the following exercises, divide.

t 6 3 t ÷ t 2 9 t 5

2 t t 3 5 t 9

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v 5 11 v ÷ v 2 25 v 11

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10 + w w 8 ÷ 100 w 2 8 w

1 10 w

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7 + x x 6 ÷ 49 x x + 6 2

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

3 y 2 ( y + 6 ) ( y + 7 ) ( y 7 ) ( y 2 + 6 )

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24 z 2 2 z 8 ÷ 4 z 28 z 2 11 z + 28

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

a ( a 5 ) a 6

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

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5 c 2 + 9 c + 2 c 2 25 ÷ 3 c 2 14 c 5 c 2 + 10 c + 25

( c + 2 ) ( c + 2 ) ( c 2 ) ( c 3 )

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

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6 m 2 2 m 10 9 m 2 ÷ 6 m 2 + 29 m 20 m 2 6 m + 9

( m 2 ) ( m 3 ) ( 3 + m ) ( m + 4 )

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

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

3 s s + 4

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r 2 9 15 ÷ r 3 27 5 r 2 + 15 r + 45

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p 3 + q 3 3 p 2 + 3 p q + 3 q 2 ÷ p 2 q 2 12

4 ( p 2 p q + q 2 ) ( p q ) ( p 2 + p q + q 2 )

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

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t 2 9 2 t ÷ ( t 2 6 t + 9 )

t + 3 2 t ( t 3 )

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

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

y + 3 z 2 y ( 2 y 1 )

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2 m 2 98 n 2 2 m + 6 ÷ ( m 2 7 m n )

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

2 a 7 5

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

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

3 ( 3 c 5 )

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

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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

4 ( m + 8 ) ( m + 7 ) 3 ( m 4 ) ( m + 2 )

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

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

( 4 p + 1 ) ( p 7 ) 3 p ( p + 9 ) ( p 1 )

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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

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Everyday math

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

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

w ( w 1 ) ( w + m ) ( w + m 1 )
2 21

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Area of a triangle The area of a triangle with base b and height h is b h 2 . 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 9 b h 2 . Calculate how the area of the new triangle compares to the area of the original triangle by dividing 9 b h 2 by b h 2 .

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Writing exercises

  1. Multiply 7 4 · 9 10 and explain all your steps.
  2. Multiply n n 3 · 9 n + 3 and explain all your steps.
  3. Evaluate your answer to part (b) when n = 7 . Did you get the same answer you got in part (a)? Why or why not?

Answers will vary.

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  1. Divide 24 5 ÷ 6 and explain all your steps.
  2. Divide x 2 1 x ÷ ( x + 1 ) and explain all your steps.
  3. Evaluate your answer to part (b) when x = 5 . Did you get the same answer you got in part (a)? Why or why not?
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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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Source:  OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2
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