ALGEBRA 1 LESSON 9 8 Factoring by Grouping

ALGEBRA 1 LESSON 9 -8 Factoring by Grouping (For help, go to Lessons 9 -2 and 9 -3. ) Find the GCF of the terms of each polynomial. 1. 6 y 2 + 12 y – 4 2. 9 r 3 + 15 r 2 + 21 r 3. 30 h 3 – 25 h 2 – 40 h 4. 16 m 3 – 12 m 2 – 36 m Find each product. 5. (v + 3)(v 2 + 5) 6. (2 q 2 – 4)(q – 5) 7. (2 t – 5)(3 t + 4) 8. (4 x – 1)(x 2 + 2 x + 3) 5 -10

ALGEBRA 1 LESSON 9 -8 Factoring by Grouping Solutions 1. 6 y 2 + 12 y – 4 6 y 2 = 2 • 3 • y; 12 y = 2 • 3 • y; 4 = 2 • 2; GCF = 2 3. 30 h 3 – 25 h 2 – 40 h 3 = 2 • 3 • 5 • h • h; 25 h 2 = 5 • h • h; 40 h = 2 • 2 • 5 • h; GCF = 5 h 2. 9 r 3 + 15 r 2 + 21 r 9 r 3 = 3 • r • r; 15 r 2 = 3 • 5 • r; 21 r = 3 • 7 • r; GCF = 3 r 4. 16 m 3 – 12 m 2 – 36 m 16 m 3 = 2 • 2 • m • m; 12 m 2 = 2 • 3 • m; 36 m = 2 • 3 • m; GCF = 2 • m = 4 m 5. (v + 3)(v 2 + 5) = (v)(v 2) + (v)(5) + (3)(v 2) + (3)(5) = v 3 + 5 v + 3 v 2 + 15 = v 3 + 3 v 2 + 5 v + 15 5 -10

ALGEBRA 1 LESSON 9 -8 Factoring by Grouping Solutions (continued) 6. (2 q 2 – 4)(q – 5) = (2 q 2)(q) + (2 q 2)(– 5) + (– 4)(q) + (– 4)(– 5) = 2 q 3 – 10 q 2 – 4 q + 20 7. (2 t – 5)(3 t + 4) = (2 t)(3 t) + (2 t)(4) + (– 5)(3 t) + (– 5)(4) = 6 t 2 + 8 t – 15 t – 20 = 6 t 2 – 7 t – 20 8. (4 x – 1)(x 2 + 2 x + 3) = (4 x)(x 2) + (4 x)(2 x) + (4 x)(3) + (– 1)(x 2) + (– 1)(2 x) + (– 1)(3) = 4 x 3 + 8 x 2 + 12 x – x 2 – 2 x – 3 = 4 x 3 + (8 – 1)x 2 + (12 – 2)x – 3 = 4 x 3 + 7 x 2 + 10 x – 3 5 -10

ALGEBRA 1 LESSON 9 -8 Factoring by Grouping Factor 6 x 3 + 3 x 2 – 4 x – 2 = 3 x 2(2 x + 1) – 2(2 x + 1) = (2 x + 1)(3 x 2 – 2) Factor the GCF from each group of two terms. Factor out (2 x + 1). Check: 6 x 3 + 3 x 2 – 4 x – 2 (2 x + 1)(3 x 2 – 2) = 6 x 3 – 4 x + 3 x 2 – 2 Use FOIL. = 6 x 3 + 3 x 2 – 4 x – 2 Write in standard form. 5 -10

ALGEBRA 1 LESSON 9 -8 Factoring by Grouping Factor 8 t 4 + 12 t 3 + 16 t + 24 = 4(2 t 4 + 3 t 3 + 4 t + 6) Factor out the GCF, 4. = 4[t 3(2 t + 3) + 2(2 t + 3)] Factor by grouping. = 4(2 t + 3)(t 3 + 2) Factor again. 5 -10

ALGEBRA 1 LESSON 9 -8 Factoring by Grouping Factor 24 h 2 + 10 h – 6. Step 1: 24 h 2 + 10 h – 6 = 2(12 h 2 + 5 h – 3) Factor out the GCF, 2. Step 2: 12 • – 3 = – 36 Find the product ac. Step 3: Factors – 2(18) = – 36 – 3(12) = – 36 – 4(9) = – 36 Sum – 2 + 18 = 16 – 3 + 12 = 9 – 4 + 9 = 5 Find two factors of ac that have a sum b. Use mental math to determine a good place to start. Step 4: 12 h 2 – 4 h + 9 h – 3 Rewrite the trinomial. Step 5: 4 h(3 h – 1) + 3(3 h – 1) Factor by grouping. Factor again. (4 h + 3)(3 h – 1) 24 h 2 + 10 h – 6 = 2(4 h + 3)(3 h – 1) 5 -10 Include the GCF in your final answer.

ALGEBRA 1 LESSON 9 -8 Factoring by Grouping A rectangular prism has a volume of 36 x 3 + 51 x 2 + 18 x. Factor to find the possible expressions for the length, width, and height of the prism. Factor 36 x 3 + 51 x 2 + 18 x. Step 1: 3 x(12 x 2 + 17 x + 6) Factor out the GCF, 3 x. Step 2: 12 • 6 = 72 Step 3: Factors 4 • 18 6 • 12 8 • 9 Find the product ac. Sum 4 + 18 = 22 6 + 12 = 18 8 + 9 = 17 5 -10 Find two factors of ac that have sum b. Use mental math to determine a good place to start.

ALGEBRA 1 LESSON 9 -8 Factoring by Grouping (continued) Step 4: 3 x(12 x 2 + 8 x + 9 x + 6) Rewrite the trinomial. Step 5: 3 x[4 x(3 x + 2) + 3(3 x + 2)] Factor by grouping. 3 x(4 x + 3)(3 x + 2) Factor again. The possible dimensions of the prism are 3 x, (4 x + 3), and (3 x + 2). 5 -10

ALGEBRA 1 LESSON 9 -8 Factoring by Grouping Factor each expression. 1. 10 p 3 – 25 p 2 + 4 p – 10 (5 p 2 + 2)(2 p – 5) 2. 36 x 4 – 48 x 3 + 9 x 2 – 12 x 3 x(4 x 2 + 1)(3 x – 4) 3. 16 a 3 – 24 a 2 + 12 a – 18 2(4 a 2 + 3)(2 a – 3) 5 -10