Do Now Pass out calculators. Have your homework out ready to check.

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Do Now Pass out calculators. Have your homework out ready to check. Solve the problem below: Divide -12 + 4y2 by -1 + 2y.

Objective: To multiply and divide rational expressions.

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 3 12 x4 = Factor and divide out common factors. 1 3 = Simplify.

GUIDED PRACTICE for Example 1 Find the product 2y3 5y 15y3 8y5 1. 3 4 ANSWER 2. 7z2 4z3 z3 14z ANSWER z 8

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 3x2 + 3x 4x2 – 24x + 36 x2 – 4x + 3 x2 – x = (3x2 + 3x) (x2 – 4x + 3) (4x2 – 24x + 36)(x2 – x) Multiply numerators and denominators. = 3x(x + 1)(x – 3)(x – 1) 4x(x – 3)(x – 3)(x – 1) Factor and divide out common factors. = 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). 5x x2 + 5x + 6 (x + 3). 5x x2 + 5x + 6 (x + 3) = 1 Rewrite polynomial as a fraction. = 5x(x + 3) x2 + 5x + 6 Multiply numerators and denominators. = 5x(x + 3) (x + 3) (x + 2) Factor and divide out common factor. = 5x x + 2 Simplify.

GUIDED PRACTICE for Examples 2 and 3 x2 + x – 2 x2 + x 2x2 + 2x 5x2 –15x +10 3. Find the product 2(x + 1) 5(x – 2) ANSWER Find the product 2w2 w2 – 7w + 12 (w – 4). 4. 2w2 w – 3 ANSWER

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 7x2 – 7x x2 + 2x – 3 x + 1 x2 – 7x – 8 7x2 – 7x x2 + 2x – 3 x + 1 = x2 – 7x – 8 Multiply by multiplicative inverse. = (7x2 – 7x) (x2 – 7x – 8) (x + 1) (x2 + 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.

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). 2x2 + 16x + 24 3x2 (x +6) = 2x2 + 16x + 24 3x2 (x +6). = 1 Rewrite polynomial as fraction. 2x2 + 16x + 24 3x2 = 1 (x +6). Multiply by multiplicative inverse. 2x2 + 16x + 24 3x2 = (x +6). Multiply numerators and denominators. = 2(x + 2)(x + 6) 3x2(x + 6) Factor and divide out common factor. = 2(x + 2) 3x2 Simplify.

GUIDED PRACTICE for Examples 4 and 5 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