Warm-Up Exercises Perform the operation. 1. x 2 + 15x + 36 x 2 – 144 2. 5x5x x 2 – 6x + 9 · x 2 + 4x – 21 x 2 + 7x ANSWERS x + 3 x – 12 ANSWERS 5 x – 3.

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Presentation transcript:

Warm-Up Exercises Perform the operation. 1. x x + 36 x 2 – x5x x 2 – 6x + 9 · x 2 + 4x – 21 x 2 + 7x ANSWERS x + 3 x – 12 ANSWERS 5 x – 3

Warm-Up Exercises EXAMPLE 1 Multiply rational expressions Multiply: x 2 + x – 20 3x3x 3x –3x 2 x 2 + 4x – 5 x 2 + x – 20 3x3x 3x –3x 2 x 2 + 4x – 5 3x(1– x) (x –1)(x +5) = (x + 5)(x – 4) 3x3x Factor numerators and denominators. 3x(1– x)(x + 5)(x – 4) = (x –1)(x + 5)(3x) Multiply numerators and denominators. 3x(–1)(x – 1)(x + 5)(x – 4) = (x – 1)(x + 5)(3x) Rewrite 1– x as (– 1)(x – 1). 3x(–1)(x – 1)(x + 5)(x – 4) = (x – 1)(x + 5)(3x) Divide out common factors. SOLUTION

Warm-Up Exercises EXAMPLE 1 Multiply rational expressions = (–1)(x – 4) Simplify. = –x + 4 Multiply. ANSWER –x + 4

Warm-Up Exercises EXAMPLE 2 Multiply a rational expression by a polynomial Multiply: x + 2 x 3 – 27 (x 2 + 3x + 9) x + 2 x 3 – 27 (x 2 + 3x + 9) Write polynomial as a rational expression. = x + 2 x 3 – 27 x 2 + 3x (x + 2)(x 2 + 3x + 9) (x – 3)(x 2 + 3x + 9) = Factor denominator. (x + 2)(x 2 + 3x + 9) (x – 3)(x 2 + 3x + 9) = Divide out common factors. = x + 2 x – 3 Simplified form SOLUTION ANSWER x + 2 x – 3

Warm-Up Exercises GUIDED PRACTICE Multiply the expressions. Simplify the result. 3x 5 y 2 8xy 6xy 2 9x3y9x3y 1.1. x2y2x2y2 4 ANSWER 2. x + 3 2x22x2 2x 2 – 10x x 2 – 25 x + 3 x(x + 5) ANSWER x (x 2 +x + 1) x 3 – 1 x + 5 x – 1 ANSWER

Warm-Up Exercises EXAMPLE 3 Divide rational expressions Divide : 7x7x 2x – 10 x 2 – 6x x 2 – 11x x7x 2x – 10 x 2 – 6x x 2 – 11x x7x 2x – 10 x 2 – 6x x 2 – 11x + 30 = Multiply by reciprocal. 7x7x 2(x – 5) = (x – 5)(x – 6) x(x – 6) = 7x(x – 5)(x – 6) 2(x – 5)(x)(x – 6) Divide out common factors. Factor. 7 2 = Simplified form SOLUTION ANSWER 7 2

Warm-Up Exercises EXAMPLE 4 Divide a rational expression by a polynomial Divide : 6x 2 + x – 15 4x24x2 (3x 2 + 5x) 6x 2 + x – 15 4x24x2 (3x 2 + 5x) 6x 2 + x – 15 4x24x2 3x 2 + 5x = 1 Multiply by reciprocal. (3x + 5)(2x – 3) 4x24x2 = x(3x + 5) 1 Factor. Divide out common factors. Simplified form 2x – 3 4x34x3 = (3x + 5)(2x – 3) = 4x 2 (x)(3x + 5) SOLUTION ANSWER 2x – 3 4x34x3

Warm-Up Exercises GUIDED PRACTICE Divide the expressions. Simplify the result. 4x4x 5x – 20 x 2 – 2x x 2 – 6x ANSWER 2x 2 + 3x – 5 6x6x (2x 2 + 5x) 5. x – 1 6x26x2 ANSWER

Warm-Up Exercises 6. x 3 – 1 2x 2 + x – 3 (x 2 + x + 1) ANSWERS 1 2x + 3 GUIDED PRACTICE