8 4 Multiply Divide Rational Expressions Factor Simplify

8 4 Multiply & Divide Rational Expressions Factor & Simplify

EXAMPLE 1 Simplify : Simplify a rational expression x 2 – 2 x – 15 x 2 – 9 SOLUTION x 2 – 2 x – 15 x 2 – 9 ANSWER (x +3)(x – 5) = (x +3)(x – 3) Factor numerator and denominator. (x +3)(x – 5) = (x +3)(x – 3) Divide out common factor. x– 5 = x– 3 Simplified form x– 5 x– 3

for Examples 1 and 2 GUIDED PRACTICE Simplify the expression, if possible. 1. 2(x + 1)(x + 3) SOLUTION 2(x + 1)(x + 3) 2(x +1) = (x +1)(x + 3) = ANSWER 2 x+3 Divide out common factor. Simplified form

for Examples 1 and 2 GUIDED PRACTICE 2. 40 x + 20 10 x + 30 SOLUTION 40 x + 20 10 x + 30 = 20(2 x +1) 10(x + 3) 20(2 x +1) = 10(x + 3) = ANSWER 2(2 x +1) x+3 Factor numerator and denominator. Divide out common factor. Simplified form

for Examples 1 and 2 GUIDED PRACTICE 3. 4 x(x + 2) SOLUTION 4 x(x + 2) ANSWER Simplified form 4 x(x + 2)

GUIDED PRACTICE 4. for Examples 1 and 2 x+4 x 2 – 16 SOLUTION x+4 x 2 – 16 (x + 4) = (x + 4)(x – 4) 1 = x– 4 ANSWER 1 x– 4 Factor numerator and denominator. Divide out common factor. Simplified form

for Examples 1 and 2 GUIDED PRACTICE 5. x 2 – 2 x – 3 x 2 – x – 6 SOLUTION x 2 – 2 x – 3 x 2 – x – 6 (x – 3)(x + 1) = (x – 3)(x + 2) Factor numerator and denominator. (x – 3)(x + 1) = (x – 3)(x + 2) Divide out common factor. = ANSWER x+1 x+2 Simplified form

for Examples 1 and 2 GUIDED PRACTICE 2 x 2 + 10 x 6. 3 x 2 + 16 x + 5 SOLUTION 2 x 2 + 10 x 3 x 2 + 16 x + 5 ANSWER 2 x(x + 5) = (3 x + 1)(x + 5) Factor numerator and denominator. = 2 x(x + 5) (3 x + 1)(x + 5) Divide out common factor. = 2 x 3 x + 1 Simplified form 2 x 3 x + 1

EXAMPLE 3 Standardized Test Practice SOLUTION 8 x 3 y 2 x y 2 7 x 4 y 3 56 x 7 y 4 = 4 y 8 xy 3 = Multiply numerators and denominators. 8 7 x x 6 y 3 y 8 x y 3 = 7 x 6 y ANSWER The correct answer is B. Factor and divide out common factors. Simplified form

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

EXAMPLE 5 Multiply a rational expression by a polynomial x+2 x 3 – 27 Multiply: (x 2 + 3 x + 9) SOLUTION = = x+2 x 3 – 27 (x 2 + 3 x + 9) x 2 + 3 x + 9 1 (x + 2)(x 2 + 3 x + 9) (x – 3)(x 2 + 3 x + 9) x+2 x– 3 ANSWER Write polynomial as a rational expression. Factor denominator. Divide out common factors. Simplified form x+2 x– 3

for Examples 3, 4 and 5 GUIDED PRACTICE Multiply the expressions. Simplify the result. 8. 3 x 5 y 2 8 xy 6 xy 2 9 x 3 y SOLUTION 3 x 5 y 2 2 xy 6 xy 2 9 x 3 y = = = 18 x 6 y 4 Multiply numerators and denominators. 72 x 4 y 2 18 x 4 y 2 x 2 y 2 18 4 x 2 y 2 4 x 4 y 2 Factor and divide out common factors. Simplified form

for Examples 3, 4 and 5 GUIDED PRACTICE 9. 2 x 2 – 10 x x 2– 25 x+3 2 x 2 SOLUTION = 2 x 2 – 10 x x 2– 25 x+3 2 x 2 2 x(x – 5)(x +5) x+3 2 x (x) Factor numerators and denominators. = 2 x(x – 5) (x + 3) (x – 5)(x + 5)2 x (x) Multiply numerators and denominators. = 2 x(x – 5) (x + 3) (x – 5)(x + 5)2 x (x) Divide out common factors. = x+3 x(x + 5) Simplified form

GUIDED PRACTICE 10. x+5 x 3– 1 for Examples 3, 4 and 5 x 2 +x + 1 SOLUTION x+5 x 3– 1 = = x 2 +x + 1 x+5 (x – 1) (x 2 +x + 1) (x + 5) (x 2 +x + 1) (x – 1) (x 2 +x + 1) x+5 x– 1 x 2 +x + 1 1 Factor denominators. Multiply numerators and denominators. Divide out common factors. Simplified form

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

EXAMPLE 7 Divide : Divide a rational expression by a polynomial 6 x 2 + x – 15 4 x 2 (3 x 2 + 5 x) SOLUTION 6 x 2 + x – 15 4 x 2 6 x 2 + x – 15 = 4 x 2 (3 x + 5)(2 x – 3) = 4 x 2 = = (3 x 2 + 5 x) 1 3 x 2 + 5 x 1 x(3 x + 5)(2 x – 3) Factor. Divide out common factors. 4 x 2(x)(3 x + 5) 2 x – 3 4 x 3 ANSWER Multiply by reciprocal. Simplified form 2 x – 3 4 x 3

for Examples 6 and 7 GUIDED PRACTICE Divide the expressions. Simplify the result. x 2 – 2 x x 2 – 6 x + 8 4 x 5 x – 20 11. SOLUTION 4 x 5 x – 20 x 2 – 2 x x 2 – 6 x + 8 4 x = 5 x – 20 x 2 – 6 x + 8 x 2 – 2 x = = = 4(x)(x – 4)(x – 2) 5(x – 4)(x)(x – 2) 4 5 Multiply by reciprocal. Factor. Divide out common factors. Simplified form

GUIDED PRACTICE 2 x 2 + 3 x – 5 6 x 12. for Examples 6 and 7 (2 x 2 + 5 x) SOLUTION 2 x 2 + 3 x – 5 6 x = = 2 x 2 + 3 x – 5 6 x (2 x + 5)(x – 1) 6 x(x)(2 x + 5) x– 1 6 x 2 (2 x 2 + 5 x) 1 (2 x 2 + 5 x) Multiply by reciprocal. Factor. Divide out common factors. Simplified form