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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 Multiply rational expressions involving monomials Find the product 2x2 3x 6x2 12x3 2x2 3x 6x2 12x3 = (2x2)(6x2) (3x)(12x3) Multiply numerators and denominators. 12x4 36x4 = Product of powers property 12 x4 x4 = Factor and divide out common factors. 1 3 = Simplify.

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**Multiply rational expressions involving polynomials**

EXAMPLE 2 Multiply rational expressions involving polynomials 3x2 + 3x 4x2 – 24x + 36 x2 – 4x + 3 x2 – x Find the product Multiply numerators and denominators. = (3x2 + 3x) (x2 – 4x + 3) (4x2 – 24x + 36)(x2 – x) Factor and divide out common factors. = 3x(x + 1)(x – 3)(x – 1) 4(x2 – 6x + 9)x(x – 1) = 3x(x + 1)(x – 3)(x – 1) 4x(x – 3)(x – 3)(x – 1) = 3(x + 1) 4(x – 3) Simplify.

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**Multiply a rational expression by a polynomial**

EXAMPLE 3 Multiply a rational expression by a polynomial Find the product 5x x2 + 5x + 6 (x + 3). Rewrite polynomial as a fraction. 5x x2 + 5x + 6 (x + 3) = 1 Multiply numerators and denominators. = 5x(x + 3) x2 + 5x + 6 Factor and divide out common factor. = 5x(x + 3) (x + 3) (x + 2) = 5x x + 2 Simplify.

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**Divide rational expressions involving polynomials**

EXAMPLE 4 Divide rational expressions involving polynomials Find the quotient 7x2 – 7x x2 + 2x – 3 x + 1 x2 – 7x – 8 Multiply by multiplicative inverse. 7x2 – 7x x2 + 2x – 3 x + 1 = x2 – 7x – 8 Multiply numerators and denominators. = (7x2 – 7x) (x2 – 7x – 8) (x + 1) (x2 + 2x – 3) Factor and divide out common factors. = 7x(x – 1)(x – 8)(x + 1) (x + 3)(x – 1)(x + 1) Simplify. = 7x(x – 8) x + 3

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**Divide a rational expression by a polynomial**

EXAMPLE 5 Divide a rational expression by a polynomial Find the quotient 2x2 + 16x + 24 3x2 (x +6). Rewrite polynomial as fraction. 2x2 + 16x + 24 3x2 (x +6). = 1 Multiply by multiplicative inverse. 2x2 + 16x + 24 3x2 = 1 (x +6). Multiply numerators and denominators. 2x2 + 16x + 24 3x2 = (x +6). Factor and divide out common factor. = 2(x + 2)(x + 6) 3x2(x + 6) = 2(x + 2) 3x2 Simplify.

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TRY THESE Find the product 2y3 5y 15y3 8y5 1. 3 4 ANSWER ANSWER z 8 2. 7z2 4z3 z3 14z x2 + x – 2 x2 + x 2x2 + 2x 5x2 –15x +10 3. 2(x + 1) 5(x – 2) ANSWER 2w2 w2 – 7w + 12 (w – 4). 4. 2w2 w – 3 ANSWER

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Find the quotient. 5. m2 – 4 2m2 + 4m 6m – 3m2 4m + 44 2 (m +11) 3m2 – ANSWER 6. n2 – 6n + 9 12n n – 3 ANSWER 12n n – 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.

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.

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