11.4 Multiply and Divide Rational Expressions

SIMPLIFYING RATIONAL EXPRESSIONS

Step 1: Factor numerator and denominator

“when in doubt, write it out!!”

Step 2: Cross out common factors

Step 3: Simplified form

SimplifyAssume the denominator cannot equal zero.

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=a4 2 − b( )a3 2 − b( )

=a4 2 − b( )a3 2 − b( )

=a4

a3 =a⋅ a⋅ a⋅ a

a⋅ a⋅ a= a

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2a4 − ba4

2a3 − a3b

Example 1Simplify a rational expressionx² + 7x + 10 x² - 4Step 1: Factor numerator and denominator(x² + 7x + 10) = (x+2)(x+5) (x² - 4) (x+2)(x-2)Step 2: Cross out common factor(x+2)(x+5)(x+2)(x-2)

Step 3: Simplified Form

x + 5

x - 2

CheckpointSimplify the expression

1. x² - 2x – 15

x² + 4x + 3

(x – 5)(x + 3)

(x + 1)(x + 3)

x – 5

x + 1

Multiply the fractionsReduce before multiply.

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=1

6x 2y€

8x

21y 3 ⋅7y 2

16x 3

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2⋅ 2⋅ 2⋅ x⋅ 7⋅ y⋅ y

7⋅ 3⋅ y⋅ y⋅ y⋅ 2⋅ 2⋅ 2⋅ 2⋅ x⋅ x⋅ x

Multiply the fractionsReduce before multiply.

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=2ac 3

3b2€

5a4c

12b⋅

24bc 2

15a3b2

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5⋅ a⋅ a⋅ a⋅ a⋅ c⋅ 2⋅ 2⋅ 2⋅ 3⋅ b⋅ c⋅ c

2⋅ 3⋅ 2⋅ b⋅ 3⋅ 5⋅ a⋅ a⋅ a⋅ b⋅ b

Example 2Multiply rational expressions

Step 1: Factor and Multiply

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2x 2 + 4 x

x 2 − 4x −12⋅

x 2 − 9x +18

2x

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2x(x + 2)(x − 3)(x − 6)

(x + 2)(x − 6)2x

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x − 3

CheckpointMultiply the expression6x² + 18x x² - x – 2x² + x – 6 * x² - 7x – 8

6x(x + 3)(x-2)(x+1)(x+3)(x-2)(x-8)(x+1) 6x x-8

More Examples Multiply the expressions. Simplify the result.

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3x 5y 2

8xy•

6xy 2

9x 3y

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2x 2 −10x

x 2 − 25•

x + 3

2x 2

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x 2y 2

4

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x + 3

x 2 + 5x

Divide the Rational ExpressionsYou can only Reduce when Multiplying

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=4dx€

10px 2

3c 2d÷

5px

6c 2d2

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2⋅ 5⋅ p⋅ x⋅ x⋅ 2⋅ 3⋅ c⋅ c⋅ d⋅ d

3⋅ c⋅ c⋅ d⋅ 5⋅ p⋅ x€

10px 2

3c 2d⋅

6c 2d2

5 px

Example 4Divide rational expressions

Step 1: Multiply by reciprocal 3 x² + 6x – 7x+7 * 8x² - 8x Step 2: Factor and Multiply 3 (x+7)(x-1)(x+7)(8x)(x-1)

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3

x + 7÷

8x 2 − 8x

x 2 + 6x − 7

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3

8x

More Examples Divide each expression. Simplify the result.

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4x

5x − 20÷

x 2 − 2x

x 2 − 6x + 8

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2x 2 + 3x − 5

6x÷ 2x 2 + 5x( )

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4

5

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(x −1)(2x + 5)

6x⋅

1

x(2x + 5)

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x −1

6x 2

CheckpointDivide the expression3.

(x – 5) (2x² - 5x + 2) (9x² - 18x)(2x² - 11x + 5)(x – 5)(2x-1)(x-2)9x(x–2)(2x-1)(x-5) 1_9x

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x − 5

9x 2 −18x÷

2x 2 −11x + 5

2x 2 − 5x + 2