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2.3 Part 1 Factoring 10/29/2012

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What is Factoring? It is finding two or more numbers or algebraic expressions, that when multiplied together produce a given product. Ex. Factor 6: 2 3 Factor 2x 2 + 4: 2 (x 2 +2) Factor x 2 +5x + 6: (x+2)(x+3)

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Type 1 Problems Factoring Quadratic equations in Standard form y = ax 2 + bx +c when a = 1 and when a > 1

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The Big “X” method c b Think of 2 numbers that Multiply to c and Add to b #1#1 #2#2 add multiply Answer: (x ± # 1 ) (x ± # 2 ) Factor: x 2 + bx + c Note: a = 1

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15 8 Think of 2 numbers that Multiply to 15 and Add to 8 3 x 5 = 15 3 + 5 = 8 3 5 Answer: (x + 3) (x + 5) Factor: x 2 + 8x + 15 c b #1#1 #2#2 add multiply

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8 -6 Think of 2 numbers that Multiply to 8 and Add to -6 -4 x -2 = 8 -4 + -2 = -6 -4 -2 Answer: (x - 4) (x - 2) To check: Foil (x – 4)(x – 2) and see if you get x 2 -6x+8 Factor: x 2 - 6x + 8 c b #1#1 #2#2 add multiply

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-9 8 Think of 2 numbers that Multiply to -9 and Add to 8 9 x -1 = -9 9 + -1 = 8 -9 8 Answer: (x - 9) (x + 8) Factor: x 2 + 8x - 9 c b #1#1 #2#2 add multiply

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The Big “X” method acac b Think of 2 numbers that Multiply to ac and Add to b #1#1 #2#2 add multiply Answer: Write the simplified answers in the 2 ( ) as binomials. Top # is coefficient of x and bottom # is the 2 nd term Factor: ax 2 + bx + c Note: a > 1 a a Simplify like a fraction if needed

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32 = 6 7 Think of 2 numbers that Multiply to 6 and Add to 7 6 x 1 = 6 6 + 1 = 7 6 1 Answer: (x + 2) (3x + 1) Factor: 3x 2 + 7x + 2 ac b #1#1 #2#2 add multiply 3 3 Simplify like a fraction. ÷ by 3 2 1 a a

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4(-9) = -36 -16 Think of 2 numbers that Multiply to -36 and Add to -16 -18 x 2 = -36 -18 + 2 = -16 -18 2 Answer: (2x - 9) (2x + 1) Factor: 4x 2 - 16x - 9 ac b #1#1 #2#2 add multiply 44 Simplify like a fraction. ÷ by 2 -9 2 a a 1 2 Simplify like a fraction. ÷ by 2

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Type 2 Problems Factoring Quadratic equations written as Difference of 2 Squares.

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Difference of Two Squares Pattern (a + b) (a – b) = a 2 – b 2 In reverse, a 2 – b 2 gives you (a + b) (a – b) Examples: 1. x 2 – 4 = x 2 – 2 2 = (x + 2) (x – 2) 2. x 2 – 144 =(x + 12) (x – 12) 3. 4x 2 – 25 = (2x + 5) (2x – 5)

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If you can’t remember that, you can still use the big X method. Factor: x 2 – 4 -4 0 Think of 2 numbers that Multiply to -4 and Add to 0 2 x -2 = -4 2 + -2 = 0 2 -2 Answer: (x + 2) (x - 2) Ex. x 2 + 0x – 4

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Ex. x 2 – 144 -144 0 Think of 2 numbers that Multiply to -144 and Add to 0 12 x -12 = -144 12 + -12 = 0 12 -12 Answer: (x + 12) (x - 12) x 2 + 0x – 144

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4(-25) = -100 0 Think of 2 numbers that Multiply to -100 and Add to 0 -10 x 10 = -100 -10 + 10 = 0 -10 10 Answer: (2x - 5) (2x + 5) Factor: 4x 2 - 25 44 Simplify like a fraction. ÷ by 2 -5 2 5 2 Simplify like a fraction. ÷ by 2 4x 2 + 0x - 25

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Type 3 Problems Factoring Quadratic equations by taking out the Greatest Common Factor

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Factor y = x 2 – 6x 1. Find the GCF. GCF = x 2. Factor the GCF out. Think reverse “distributive prop.” y = x (x – 6)

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Factor y = -8x 2 + 18 1. Find the GCF. GCF = -2 Why -2 and not 2 you ask? Wait for the next step. 2. Factor the GCF out. y = -2 (x 2 - 9) Answer: So we can have the difference of 2 squares pattern y = 2 (-x 2 + 9) Not Difference of 2 Squares 3. Factor what’s in the ( ) since it follows the difference of 2 square pattern. y = -2(x – 3)(x + 3)

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