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11.5 Multiplying and Dividing Rational Expressions Multiply and divide rational expressions. Rules for multiplying and dividing rational expressions are the same as for multiplying and dividing fractions.

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Multiply rational expressions involving monomials EXAMPLE 1 Find the product 2x22x2 3x 6x 2 12x 3 Multiply numerators and denominators. Product of powers property Factor and divide out common factors. Simplify. 1 3 = 12 x x 4 = 12x 4 36x 4 = 2x22x2 3x 6x26x2 12x 3 = (2x 2 )(6x 2 ) (3x)(12x 3 )

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EXAMPLE 2 Multiply rational expressions involving polynomials Find the product 3x 2 + 3x 4x 2 – 24x + 36 x 2 – 4x + 3 x 2 – x Multiply numerators and denominators. = (3x 2 + 3x) (x 2 – 4x + 3) (4x 2 – 24x + 36)(x 2 – x) Factor and divide out common factors. = 3(x + 1) 4(x – 3) Simplify. = 3x(x + 1)(x – 3)(x – 1) 4x(x – 3)(x – 3)(x – 1) = 3x(x + 1)(x – 3)(x – 1) 4(x 2 – 6x + 9)x(x – 1)

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EXAMPLE 3 Multiply a rational expression by a polynomial Find the product 5x5x x 2 + 5x + 6 (x + 3). 5x5x x 2 + 5x + 6 (x + 3) = 1 Rewrite polynomial as a fraction. = 5x(x + 3) x 2 + 5x + 6 Multiply numerators and denominators. = 5x(x + 3) (x + 3)(x + 2) Factor and divide out common factor. Simplify. = 5x5x x + 2

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EXAMPLE 4 Divide rational expressions involving polynomials Find the quotient. 7x 2 – 7x x 2 + 2x – 3 x + 1 x 2 – 7x – 8 7x 2 – 7x x 2 + 2x – 3x + 1 = x 2 – 7x – 8 Multiply by multiplicative inverse. = (7x 2 – 7x) (x 2 – 7x – 8) (x + 1)(x 2 + 2x – 3) Multiply numerators and denominators. = 7x(x – 1)(x – 8)(x + 1) (x + 3)(x – 1)(x + 1) Factor and divide out common factors. = 7x(x – 8) x + 3 Simplify.

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EXAMPLE 5 Divide a rational expression by a polynomial Find the quotient 2x x x23x2 (x +6).2x x x23x2 (x +6). = 1 Rewrite polynomial as fraction. 2x x x23x2 = 1 (x +6). Multiply by multiplicative inverse. 2x x x23x2 = (x +6). Multiply numerators and denominators. = 2(x + 2)(x + 6) 3x 2 (x + 6) Factor and divide out common factor. = 2(x + 2) 3x23x2 Simplify.

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TRY THESE Find the product 2y32y3 5y 15y 3 8y ANSWER 2. 7z27z2 4z 3 z 3 14z ANSWER z 8 3. x 2 + x – 2 x 2 + x 2x 2 + 2x 5x 2 –15x +10 2(x + 1) 5(x – 2) ANSWER 2w22w2 w 2 – 7w + 12 (w – 4). 4. 2w22w2 w – 3 ANSWER

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5. m 2 – 4 2m 2 + 4m 6m – 3m 2 4m (m +11) 3m23m2 – ANSWER 6. n 2 – 6n n n – 3 ANSWER 12n n – 3 Find the quotient.

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