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

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

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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

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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)

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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)

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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)

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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)

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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)

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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)

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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!

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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)

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