# 11.5 Multiplying and Dividing Rational Expressions

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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.

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.

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.

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.

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

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.

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

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