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( x - 3 ) · 4 x - 9 x 2 - 6 x + 9 . ( x - 3 ) 1 · 4 x - 9 ( x - 3 ) ( x - 3 ) 4 x - 9 x - 3

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- x 2 - 3 x - 2 x 2 + 8 x + 15 · 4 x + 20 x 2 + 2 x . Factor –1 from the first numerator . - ( x 2 + 3 x + 2 ) x 2 + 8 x + 15 · 4 x + 20 x 2 + 2 x Factor . - ( x + 1 ) ( x + 2 ) ( x + 3 ) ( x + 5 ) · 4 ( x + 5 ) x ( x + 2 ) Multiply . - 4 ( x + 1 ) x ( x + 3 ) = - 4 x - 1 x ( x + 3 ) or - 4 x - 1 x 2 + 3 x

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Practice set a

Perform each multiplication.

a 3 b 2 c 2 · c 5 a 5

c 3 a 2 b 2

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y - 1 y 2 + 1 · y + 1 y 2 - 1

1 y 2 + 1

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x 2 - x - 12 x 2 + 7 x + 6 · x 2 - 4 x - 5 x 2 - 9 x + 20

x + 3 x + 6

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x 2 + 6 x + 8 x 2 - 6 x + 8 · x 2 - 2 x - 8 x 2 + 2 x - 8

( x + 2 ) 2 ( x - 2 ) 2

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Division of rational expressions

To divide one rational expression by another, we first invert the divisor then multiply the two expressions. Symbolically, if we let P , Q , R , and S represent polynomials, we can write

P Q ÷ R S = P Q · S R = P · S Q · R

Sample set b

Perform the following divisions.

6 x 2 5 a ÷ 2 x 10 a 3 . Invert the divisor and multiply . 6 3 x 2 5 a · 10 2 a 3 2 2 x = 3 x · 2 a 2 1 = 6 a 2 x

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x 2 + 3 x - 10 2 x - 2 ÷ x 2 + 9 x + 20 x 2 + 3 x - 4 Invert and multiply . x 2 + 3 x - 10 2 x - 2 · x 2 + 3 x - 4 x 2 + 9 x + 20 Factor . ( x + 5 ) ( x - 2 ) 2 ( x - 1 ) · ( x + 4 ) ( x - 1 ) ( x + 5 ) ( x + 4 ) x - 2 2

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( 4 x + 7 ) ÷ 12 x + 21 x - 2 . Write  4 x + 7  as  4 x + 7 1 . 4 x + 7 1 ÷ 12 x + 21 x - 2 Invert and multiply . 4 x + 7 1 · x - 2 12 x + 21 Factor . 4 x + 7 1 · x - 2 3 ( 4 x + 7 ) = x - 2 3

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Practice set b

Perform each division.

8 m 2 n 3 a 5 b 2 ÷ 2 m 15 a 7 b 2

20 a 2 m n

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x 2 - 4 x 2 + x - 6 ÷ x 2 + x - 2 x 2 + 4 x + 3

x + 1 x - 1

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6 a 2 + 17 a + 12 3 a + 2 ÷ ( 2 a + 3 )

3 a + 4 3 a + 2

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Excercises

For the following problems, perform the multiplications and divisions.

4 a 3 5 b · 3 b 2 a

6 a 2 5

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12 a 3 7 · 28 15 a

16 a 2 5

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18 x 6 7 · 1 4 x 2

9 x 4 14

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16 x 6 y 3 15 x 2 · 25 x 4 y

20 x 5 y 2 3

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27 a 7 b 4 39 b · 13 a 4 b 2 16 a 5

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10 x 2 y 3 7 y 5 · 49 y 15 x 6

14 3 x 4 y

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22 m 3 n 4 11 m 6 n · 33 m n 4 m n 3

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- 10 p 2 q 7 a 3 b 2 · 21 a 5 b 3 2 p

15 a 2 b p q

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- 25 m 4 n 3 14 r 3 s 3 · 21 r s 4 10 m n

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21 a 4 5 b 2 ÷ 14 a 15 b 3

9 a 3 b 2

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42 x 5 16 y 4 ÷ 21 x 4 8 y 3

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39 x 2 y 2 55 p 2 ÷ 13 x 3 y 15 p 6

9 p 4 y 11 x

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14 m n 3 25 n 6 ÷ 32 m 20 m 2 n 3

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12 a 2 b 3 - 5 x y 4 ÷ 6 a 2 15 x 2

6 b 3 x y 4

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24 p 3 q 9 m n 3 ÷ 10 p q - 21 n 2

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x + 8 x + 1 · x + 2 x + 8

x + 2 x + 1

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x + 10 x - 4 · x - 4 x - 1

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2 x + 5 x + 8 · x + 8 x - 2

2 x + 5 x 2

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y + 2 2 y - 1 · 2 y - 1 y - 2

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x - 5 x - 1 ÷ x - 5 4

4 x 1

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a + 2 b a - 1 ÷ 4 a + 8 b 3 a - 3

3 4

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6 m + 2 m - 1 ÷ 4 m - 4 m - 1

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x 3 · 4 a b x

4 a b x 2

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2 a 5 ÷ 6 a 2 4 b

4 a 3 b 3

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21 m 4 n 2 ÷ 3 m n 2 7 n

49 m 3 n

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( x + 8 ) · x + 2 x + 8

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( x - 2 ) · x - 1 x - 2

x 1

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( a - 6 ) 3 · ( a + 2 ) 2 a - 6

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( b + 1 ) 4 · ( b - 7 ) 3 b + 1

( b + 1 ) 3 ( b 7 ) 3

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( b 2 + 2 ) 3 · b - 3 ( b 2 + 2 ) 2

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( x 3 - 7 ) 4 · x 2 - 1 ( x 3 - 7 ) 2

( x 3 7 ) 2 ( x + 1 ) ( x 1 )

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( x - 5 ) ÷ x - 5 x - 2

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( y - 2 ) ÷ y - 2 y - 1

( y 1 )

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( y + 6 ) 3 ÷ ( y + 6 ) 2 y - 6

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( a - 2 b ) 4 ÷ ( a - 2 b ) 2 a + b

( a 2 b ) 2 ( a + b )

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x 2 + 3 x + 2 x 2 - 4 x + 3 · x 2 - 2 x - 3 2 x + 2

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6 x - 42 x 2 - 2 x - 3 · x 2 - 1 x - 7

6 ( x 1 ) ( x 3 )

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3 a + 3 b a 2 - 4 a - 5 ÷ 9 a + 9 b a 2 - 3 a - 10

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a 2 - 4 a - 12 a 2 - 9 ÷ a 2 - 5 a - 6 a 2 + 6 a + 9

( a + 2 ) ( a + 3 ) ( a 3 ) ( a + 1 )

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b 2 - 5 b + 6 b 2 - b - 2 · b 2 - 2 b - 3 b 2 - 9 b + 20

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m 2 - 4 m + 3 m 2 + 5 m - 6 · m 2 + 4 m - 12 m 2 - 5 m + 6

1

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r 2 + 7 r + 10 r 2 - 2 r - 8 ÷ r 2 + 6 r + 5 r 2 - 3 r - 4

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2 a 2 + 7 a + 3 3 a 2 - 5 a - 2 · a 2 - 5 a + 6 a 2 + 2 a - 3

( 2 a + 1 ) ( a 6 ) ( a + 1 ) ( 3 a + 1 ) ( a 1 ) ( a 2 )

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6 x 2 + x - 2 2 x 2 + 7 x - 4 · x 2 + 2 x - 12 3 x 2 - 4 x - 4

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x 3 y - x 2 y 2 x 2 y - y 2 · x 2 - y x - x y

x ( x y ) 1 y

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4 a 3 b - 4 a 2 b 2 15 a - 10 · 3 a - 2 4 a b - 2 b 2

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x + 3 x - 4 · x - 4 x + 1 · x - 2 x + 3

x 2 x + 1

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x - 7 x + 8 · x + 1 x - 7 · x + 8 x - 2

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2 a - b a + b · a + 3 b a - 5 b · a - 5 b 2 a - b

a + 3 b a + b

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3 a ( a + 1 ) 2 a - 5 · 6 ( a - 5 ) 2 5 a + 5 · 15 a + 30 4 a - 20

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- 3 a 2 4 b · - 8 b 3 15 a

2 a b 2 5

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- 6 x 3 5 y 2 · 20 y - 2 x

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- 8 x 2 y 3 - 5 x ÷ 4 - 15 x y

6 x 2 y 4

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- 4 a 3 3 b ÷ 2 a 6 b 2

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- 3 a - 3 2 a + 2 · a 2 - 3 a + 2 a 2 - 5 a - 6

3 ( a 2 ) ( a 1 ) 2 ( a 6 ) ( a + 1 )

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x 2 - x - 2 x 2 - 3 x - 4 · - x 2 + 2 x + 3 - 4 x - 8

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- 5 x - 10 x 2 - 4 x + 3 · x 2 + 4 x + 1 x 2 + x - 2

5 ( x 2 + 4 x + 1 ) ( x 3 ) ( x 1 ) 2

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- a 2 - 2 a + 15 - 6 a - 12 ÷ a 2 - 2 a - 8 - 2 a - 10

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- b 2 - 5 b + 14 3 b - 6 ÷ - b 2 - 9 b - 14 - b + 8

( b 8 ) 3 ( b + 2 )

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3 a + 6 4 a - 24 · 6 - a 3 a + 15

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4 x + 12 x - 7 · 7 - x 2 x + 2

2 ( x + 3 ) ( x + 1 )

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- 2 b - 2 b 2 + b - 6 · - b + 2 b + 5

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3 x 2 - 6 x - 9 2 x 2 - 6 x - 4 ÷ 3 x 2 - 5 x - 2 6 x 2 - 7 x - 3

3 ( x 3 ) ( x + 1 ) ( 2 x 3 ) 2 ( x 2 3 x 2 ) ( x 2 )

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- 2 b 2 - 2 b + 4 8 b 2 - 28 b - 16 ÷ b 2 - 2 b + 1 2 b 2 - 5 b - 3

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x 2 + 4 x + 3 x 2 + 5 x + 4 ÷ ( x + 3 )

( x + 4 ) ( x 1 ) ( x + 3 ) ( x 2 4 x 3 )

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x 2 - 3 x + 2 x 2 - 4 x + 3 ÷ ( x - 3 )

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3 x 2 - 21 x + 18 x 2 + 5 x + 6 ÷ ( x + 2 )

3 ( x 6 ) ( x 1 ) ( x + 2 ) 2 ( x + 3 )

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Exercises for review

( [link] ) If a < 0 , then | a | = .

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( [link] ) Classify the polynomial 4 x y + 2 y as a monomial, binomial, or trinomial. State its degree and write the numerical coefficient of each term.

binomial; 2; 4, 2

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( [link] ) Find the product: y 2 ( 2 y 1 ) ( 2 y + 1 ) .

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( [link] ) Translate the sentence “four less than twice some number is two more than the number” into an equation.

2 x 4 = x + 2

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( [link] ) Reduce the fraction x 2 - 4 x + 4 x 2 - 4 .

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Source:  OpenStax, Elementary algebra. OpenStax CNX. May 08, 2009 Download for free at http://cnx.org/content/col10614/1.3
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