WarmUp Exercises Factor the expression 1 10 x
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Warm-Up Exercises Factor the expression. 1. 10 x – 5 x 2 ANSWER 5 x (2 – x) 2. x 2 – 2 x – 48 ANSWER (x + 6)(x – 8)
Warm-Up Exercises Factor the expression. 3. x 3 – 125 ANSWER (x – 5)(x 2 + 5 x + 25)
Warm-Up 1 Exercises EXAMPLE Simplify a rational expression 2 – 2 x – 15 x Simplify : x 2 – 9 SOLUTION x 2 – 2 x – 15 (x +3)(x – 5) = (x +3)(x – 3) x 2 – 9 ANSWER 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
Warm-Up YOU TRY Exercises for Examples 1 and 2 Simplify the expression, if possible. 1. 2(x + 1)(x + 3) ANSWER 2 x+3 2. 40 x + 20 10 x + 30 ANSWER 3. x(x 4+ 2) ANSWER 2(2 x +1) x+3 4 x(x + 2) is in simplified form
Warm-Up YOU TRY Exercises for Examples 1 and 2 Simplify the expression, if possible. 4. x+4 x 2 – 16 ANSWER 5. 1 x– 4 x 2 – 2 x – 3 x 2 – x – 6 ANSWER x+1 x+2 2 + 10 x 2 x 6. 3 x 2 + 16 x + 5 2 x ANSWER 3 x + 1
Warm-Up 2 Exercises EXAMPLE Standardized Test Practice SOLUTION 8 x 3 y 7 x 4 y 3= 56 x 7 y 4 8 xy 3 2 x y 2 4 y 8 7 x x 6 y 3 y = 8 x y 3 = 7 x 6 y Multiply numerators and denominators. Factor and divide out common factors. Simplified form ANSWER The correct answer is B.
Warm-Up 3 Exercises EXAMPLE Multiply rational expressions 2 3 x – 3 x Multiply: 2 x + 4 x – 5 x 2 + x – 20 3 x SOLUTION 3 x – 3 x 2 + x – 20 x 2 + 4 x – 5 3 x (x + 5)(x – 4) 3 x(1– x) = (x – 1)(x +5) 3 x Factor numerators and denominators. = 3 x(1– x)(x + 5)(x – 4) (x – 1)(x + 5)(3 x) 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.
Warm-Up 3 Exercises EXAMPLE Multiply rational expressions = (– 1)(x – 4) Simplify. = –x + 4 Multiply. ANSWER –x + 4
Warm-Up 4 Exercises EXAMPLE Multiply a rational expression by a polynomial Multiply: x 3 + 2 (x 2 + 3 x + 9) x – 27 SOLUTION x + 2 (x 2 + 3 x + 9) x 3 – 27 2 + 3 x + 9 x + 2 x = 3 x – 27 1 (x + 2)(x 2 + 3 x + 9) = (x – 3)(x 2 + 3 x + 9) x+2 = x– 3 x+2 ANSWER x– 3 Write polynomial as a rational expression. Factor denominator. Divide out common factors. Simplified form
Warm-Up YOU TRY Exercises for Examples 3 and 4 Multiply the expressions. Simplify the result. 7. 3 x 5 y 2 6 xy 2 8 xy 9 x 3 y ANSWER 2 – 10 x 2 x 8. x 2– 25 ANSWER 9. x 3+ 5 x – 1 ANSWER x 2 y 2 4 x+3 2 x 2 x+3 x(x + 5) (x 2 +x + 1) x+5 x– 1
Warm-Up 5 Exercises EXAMPLE Divide rational expressions 7 x Divide : 2 x – 10 x 2 – 6 x x 2 – 11 x + 30 SOLUTION 7 x x 2 – 6 x 2 x – 10 x 2 – 11 x + 30 7 x x = 2 x – 10 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) = 7 2 ANSWER 7 2 Multiply by reciprocal. Factor. Divide out common factors. Simplified form
Warm-Up 6 Exercises EXAMPLE Divide a rational expression by a polynomial 2 + x – 15 6 x Divide : 4 x 2 (3 x 2 + 5 x) SOLUTION 6 x 2 + x – 15 2 + 5 x) (3 x 2 4 x 2 + x – 15 6 x 1 = 2 3 x + 5 x 4 x 2 (3 x + 5)(2 x – 3) 1 = x(3 x + 5) 4 x 2 (3 x + 5)(2 x – 3) = 2 4 x (x)(3 x + 5) 2 x – 3 = 4 x 3 2 x – 3 ANSWER 4 x 3 Multiply by reciprocal. Factor. Divide out common factors. Simplified form
Warm-Up YOU TRY Exercises for Examples 5 and 6 Divide the expressions. Simplify the result. 2 – 2 x x 4 x 10. 5 x – 20 x 2 – 6 x + 8 ANSWER 11. 4 5 2 x 2 + 3 x – 5 6 x ANSWER x– 1 6 x 2 (2 x 2 + 5 x)
Warm-Up Exercises KEEP GOING Perform the operation. 2 + 15 x + 36 x 1. x 2 – 144 x+3 x – 12 ANSWERS 5 x 2. 2 x – 6 x + 9 ANSWERS · x 2 + 4 x – 21 x 2 + 7 x 5 x– 3
Warm-Up Exercises KEEP GOING 3– 1 x 3. 2 x 2 + x – 3 ANSWERS (x 2 + x + 1) 1 2 x + 3 4. A cube has edge length x. A rectangular prism has a square base with side length x and height x + 2. Write an expression in simplest form for the ratio of the surface area of the prism to The surface area of the cube. ANSWERS 3 x + 4 3 x
