1. Lesson 9-8 Factoring by Grouping Designed by Skip Tyler, Varina High School

2. Objective The student will be able to: use grouping to factor polynomials with four terms. Designed by Skip Tyler, Varina High School

3. Factoring Chart This chart will help you to determine which method of factoring to use. TypeNumber of Terms 1.GCF 2 or more 2.X-Box 3 3.Special Cases 2 or 3 (Perfect Squares) 4. Grouping 4

4. 1. Factor 6n 3 + 8n 2 + 3n + 4 Do you have a GCF for all 4 terms? No Group the first 2 terms and the last 2 terms. (6n 3 + 8n 2 ) + (3n + 4) Find the GCF of each group. 2n 2 (3n + 4) + 1(3n + 4) The parentheses are the same! (3n + 4)(2n 2 + 1)

5. 2. Factor 12ac + 21ad + 8bc + 14bd Do you have a GCF for all 4 terms? No Group the first 2 terms and the last 2 terms. (12ac + 21ad) + (8bc + 14bd) Find the GCF of each group. 3a (4c + 7d) + 2b(4c + 7d) The parentheses are the same! (3a + 2b)(4c + 7d)

6. 3. Factor rx + 2ry + kx + 2ky Check for a GCF: None You have 4 terms - try factoring by grouping. (rx + 2ry) + (kx + 2ky) Find the GCF of each group. r(x + 2y) + k(x + 2y) The parentheses are the same! (r + k)(x + 2y)

7. 4. Factor 2x 2 - 3xz - 2xy + 3yz Check for a GCF: None Factor by grouping. Parentheses need to match! (2x 2 - 3xz) + (- 2xy + 3yz) Find the GCF of each group. x(2x – 3z) + y(- 2x + 3z) The signs are opposite in the parentheses! Something has to change! Factor out a negative. x(2x – 3z) - y(2x - 3z) (x - y)(2x - 3z)

8. 5.Factor 150x 3 + 350x 2 + 180x +420 Check for a GCF: 10 Factor out the 10! 10(15x 3 + 35x 2 + 18x + 42) Group the first two terms and the last two terms. 10( (15x 3 + 35x 2 ) (+ 18x + 42)) Find the GCF of each group. 10( 5x 2 (3x + 7) +6(3x + 7)) Do your parentheses match? 10(5x 2 +6)(3x + 7)

9. 6. Factor 12p 2 + 16p + 5 Check for a GCF: None Factor by grouping. This one will be different, why? It’s a trinomial!!! Use the X Box method! 16 (12)(5)= 60 6 10 5 10p 6p 12p 2 6p5 2p 1 12p 2 + 16p + 5 = (2p + 1)(6p + 5)

10. Challenging Problem: 27m 3 + 45m 2 – 3m – 5 Check for a GCF: no Group the first two terms and the last two terms. (27m 3 + 45m 2 ) (– 3m – 5) Find the GCF of each group. 9m 2 (3m+5) +1(-3m-5) Do your parentheses match? NO 9m 2 (3m+5) -1(3m+5) Do your parentheses match? Yes (3m+5)(9m 2 -1)  but wait, perfect square special case (3m+5)(3m-1)(3m+1) NOW it’s completely factored!

11. Challenging Problem: 2m 3 + 16 + m + 32m 2 This problem isn’t in standard form! 2m 3 + 32m 2 + m + 16 Did your parentheses match? What are you going to have to do to get the parentheses to match? What’s different about this problem than the others? Group the 1 st two terms and the last two terms. 2m 2 (m+16) + 1 (m + 16) (2m 2 + 1)(m+16)