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8.2B Factoring by Grouping Objectives The student will be able to: use grouping to factor polynomials with four terms use the zero product property to solve equations
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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. Grouping 4
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Ex 1) 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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Ex 2) 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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Ex 3) Factor 16k 3 – 4k 2 p 2 – 28kp + 7p 3 Check for a GCF: None Factor by grouping. Keep a + between the groups. (16k 3 – 4k 2 p 2 ) + (– 28kp + 7p 3 ) Find the GCF of each group. 4k 2 (4k – p 2 ) + 7p(– 4k + p 2 ) The signs are opposite in the parentheses! Keep-change-change! 4k 2 (4k – p 2 ) – 7p(4k – p 2 ) (4k 2 – 7p)(4k – p 2 )
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Zero Product Property If a b = 0 then a=0, b=0, or both a and b equal 0.
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Ex 4) Solve (2a + 4)(a + 7) = 0 2a + 4 = 0 or a + 7 = 0 2a = – 4 or a = – 7 a = – 2 or a = – 7 {– 2, – 7}
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4 steps for solving a quadratic equation 1.Set the equation equal to 0. 2.Factor the equation. 3.Set each part equal to 0 and solve. 4.Check your answer on the calculator. Set = 0 Factor Split/Solve Check
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Ex 5) Solve: x 2 – 11x = 0 GCF = x x(x – 11) = 0 x = 0 or x – 11 = 0 x = 0 or x = 11 {0, 11} Set = 0 Factor Split/Solve Check
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Ex 6) Solve: – 24a +144 = – a 2 Put it in descending order. a 2 – 24a + 144 = 0 (a – 12) 2 = 0 a – 12 = 0 a = 12 {12} Set = 0 Factor Split/Solve Check
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Ex 7) Solve: x 3 + 2x 2 = 15x x 3 + 2x 2 – 15x = 0 x(x 2 + 2x – 15) = 0 x(x + 5)(x – 3) = 0 x = 0 or x + 5 = 0 or x – 3 = 0 {0, – 5, 3} Set = 0 Factor Split/Solve Check
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8.2B HW PG. 479 #5 – 13 ODDS, 21 – 41 ODDS (16 PROBLEMS)
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