Factoring by Grouping Find the GCF of the terms of each polynomial. ALGEBRA 1 LESSON 9-8 (For help, go to Lessons 9-2 and 9-3.) Find the GCF of the terms of each polynomial. 1. 6y2 + 12y – 4 2. 9r3 + 15r2 + 21r 3. 30h3 – 25h2 – 40h 4. 16m3 – 12m2 – 36m Find each product. 5. (v + 3)(v2 + 5) 6. (2q2 – 4)(q – 5) 7. (2t – 5)(3t + 4) 8. (4x – 1)(x2 + 2x + 3) 5-10

Factoring by Grouping Solutions 1. 6y2 + 12y – 4 2. 9r3 + 15r2 + 21r ALGEBRA 1 LESSON 9-8 Solutions 1. 6y2 + 12y – 4 2. 9r3 + 15r2 + 21r 6y2 = 2 • 3 • y • y; 9r3 = 3 • 3 • r • r • r; 12y = 2 • 2 • 3 • y; 4 = 2 • 2; 15r2 = 3 • 5 • r • r; 21r = 3 • 7 • r; GCF = 2 GCF = 3r 3. 30h3 – 25h2 – 40h 4. 16m3 – 12m2 – 36m 30h3 = 2 • 3 • 5 • h • h • h; 16m3 = 2 • 2 • 2 • 2 • m • m • m; 25h2 = 5 • 5 • h • h; 12m2 = 2 • 2 • 3 • m • m; 40h = 2 • 2 • 2 • 5 • h; 36m = 2 • 2 • 3 • 3 • m; GCF = 5h GCF = 2 • 2 • m = 4m 5. (v + 3)(v2 + 5) = (v)(v2) + (v)(5) + (3)(v2) + (3)(5) = v3 + 5v + 3v2 + 15 = v3 + 3v2 + 5v + 15 5-10

Factoring by Grouping Solutions (continued) 6. (2q2 – 4)(q – 5) ALGEBRA 1 LESSON 9-8 Solutions (continued) 6. (2q2 – 4)(q – 5) = (2q2)(q) + (2q2)(–5) + (–4)(q) + (–4)(–5) = 2q3 – 10q2 – 4q + 20 7. (2t – 5)(3t + 4) = (2t)(3t) + (2t)(4) + (–5)(3t) + (–5)(4) = 6t2 + 8t – 15t – 20 = 6t2 – 7t – 20 8. (4x – 1)(x2 + 2x + 3) = (4x)(x2) + (4x)(2x) + (4x)(3) + (–1)(x2) + (–1)(2x) + (–1)(3) = 4x3 + 8x2 + 12x – x2 – 2x – 3 = 4x3 + (8 – 1)x2 + (12 – 2)x – 3 = 4x3 + 7x2 + 10x – 3 5-10

Factoring by Grouping Factor 6x3 + 3x2 – 4x – 2. ALGEBRA 1 LESSON 9-8 Factor 6x3 + 3x2 – 4x – 2. 6x3 + 3x2 – 4x – 2 = 3x2(2x + 1) – 2(2x + 1) Factor the GCF from each group of two terms. = (2x + 1)(3x2 – 2) Factor out (2x + 1). = 6x3 – 4x + 3x2 – 2 Use FOIL. Check: 6x3 + 3x2 – 4x – 2 (2x + 1)(3x2 – 2) = 6x3 + 3x2 – 4x – 2 Write in standard form. 5-10

Factoring by Grouping Factor 8t4 + 12t3 + 16t + 24. ALGEBRA 1 LESSON 9-8 Factor 8t4 + 12t3 + 16t + 24. 8t4 + 12t3 + 16t + 24 = 4(2t4 + 3t3 + 4t + 6) Factor out the GCF, 4. = 4[t3(2t + 3) + 2(2t + 3)] Factor by grouping. = 4(2t + 3)(t3 + 2) Factor again. 5-10

Factoring by Grouping Factor 24h2 + 10h – 6. ALGEBRA 1 LESSON 9-8 Factor 24h2 + 10h – 6. Step 1: 24h2 + 10h – 6 = 2(12h2 + 5h – 3)   Factor out the GCF, 2. Step 2: 12 • –3 = –36 Find the product ac. Step 3: Factors Sum –2(18) = –36 –2 + 18 = 16 –3(12) = –36 –3 + 12 = 9 –4(9) = –36 –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:  12h2 – 4h + 9h – 3 Rewrite the trinomial. Step 5:   4h(3h – 1) + 3(3h – 1) Factor by grouping.   (4h + 3)(3h – 1) Factor again. 24h2 + 10h – 6 = 2(4h + 3)(3h – 1) Include the GCF in your final answer. 5-10

Factoring by Grouping ALGEBRA 1 LESSON 9-8 A rectangular prism has a volume of 36x3 + 51x2 + 18x. Factor to find the possible expressions for the length, width, and height of the prism. Factor 36x3 + 51x2 + 18x. Step 1: 3x(12x2 + 17x + 6) Factor out the GCF, 3x. Step 2: 12 • 6 = 72 Find the product ac. Step 3:  Factors     Sum 4 • 18 4 + 18 = 22 6 • 12 6 + 12 = 18 8 • 9 8 + 9 = 17 Find two factors of ac that have sum b. Use mental math to determine a good place to start. 5-10

Factoring by Grouping (continued) ALGEBRA 1 LESSON 9-8 (continued) Step 4: 3x(12x2 + 8x + 9x + 6) Rewrite the trinomial. Step 5: 3x[4x(3x + 2) + 3(3x + 2)] Factor by grouping. 3x(4x + 3)(3x + 2) Factor again. The possible dimensions of the prism are 3x, (4x + 3), and (3x + 2). 5-10

Factoring by Grouping Factor each expression. 1. 10p3 – 25p2 + 4p – 10 ALGEBRA 1 LESSON 9-8 Factor each expression. 1. 10p3 – 25p2 + 4p – 10 2. 36x4 – 48x3 + 9x2 – 12x 3. 16a3 – 24a2 + 12a – 18 (5p2 + 2)(2p – 5) 3x(4x2 + 1)(3x – 4) 2(4a2 + 3)(2a – 3) 5-10