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Follow these basic steps …

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Factor out the GCF.

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Count how many terms and try the following tactics. Then, go to step 3. 2 terms -- difference of 2 squares: a 2 – b 2 = (a + b)(a – b) Example:Factor 64x 4 – 9y 2 a = 8x 2 and b = 3y = (8x 2 + 3y)(8x 2 – 3y)

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2 terms -- difference of 2 cubes: a 3 – b 3 = (a - b)(a 2 + ab + b 2 ) Example:Factor 8x 3 – y 3 SOMPS = (2x- y)( SOMPS S q u a r e f i r s t t e r m O p p o s i t e s i g n M u l t i p l y P l u s S q u a r e S e c o n d t e r m 4x 2 +2xy+y2)y2)

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2 terms -- sum of 2 cubes: a 3 + b 3 = (a + b)(a 2 - ab + b 2 ) Example:Factor 64s 3 + t 6 SOMPS ** Notice that SOMPS still works here. = (4s+ t 2 )(16s 2 -4st 2 +t4)t4) NOTE: a 2 + b 2 is NOT factorable.

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3 terms -- Set up 2 ( )’s use factors of 1 st term and last term until you get a pair that works with middle terms (guess and check) Example:Factor x 2 - 7x - 18 1, 1 1,18 2,9 3,6 Signs must be different = (1x + 2)(1x – 9) Check middle terms 2x -9x -7x Matches middle term of original. Yea!

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Example:Factor 6x 2 + 17x + 12 Signs must be the same 1, 6 2,3 1,12 2,6 3,4 = (3x + 4)(2x + 3) 8x 9x 17x ** This can be exhausting (trying to pick the factors that work)! Try alternate method

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4 or more terms -- Try grouping Intro: Factor xz – xy =x(z – y) Factor (x + 2)z – (x + 2)y = (x + 2)(z – y) Example:Factor 2x + x 2 – 6y – 3xy S1: group the terms – I pick the 1 st and the 3 rd terms; I reorder. = 2x – 6y + x 2 – 3xy S2: Factor out the GCF from each pair = 2 + x (x – 3y) S3: Since (x – 3y) is the same in both terms – factor it out. = (x – 3y)(2 + x)

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Repeat steps until all factors are prime; i.e., they can’t be factored anymore.

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Factor 4x 6 – 64x 2 Step 1 – Factor out GCF = 4x 2 (x 4 – 16) Step 2 – Count how many terms 2 terms – it’s the difference of 2 squares = 4x 2 (x 2 + 4)(x 2 – 4) Step 3 – Keep repeating until all factors are prime = 4x 2 (x 2 + 4)(x + 2)(x - 2)

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this will ALWAYS work with a factorable trinomial Example:Factor 6x 2 – x - 12 ax 2 + bx + c a = 6, b = -1, c = -12 STEP 1: Write in standard form and recognize a, b, and c. STEP 2: Multiply ac. (6)(-12) = -72 STEP 3: List all factors of ac. Circle the factors that add up to b. -1,72 -2,36 -3,24 -4,18 -6,12 -8,9 1,-72 2,-36 3,-24 4,-18 6,-12 8,-9

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this will ALWAYS work with a factorable trinomial Example:Factor 6x 2 – x - 12 a = 1, b = -1, c = -12 STEP 4: Replace bx (in original) with factors. = 6x 2 – 9x + 8x - 12 STEP 5: Group 1 st two terms and last 2 terms. = 6x 2 – 9x + 8x - 12 STEP 6: Factor. = 3x + 4 (2x – 3) = (2x – 3)(3x + 4) Back to Notes

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Purpose: To factor polynomials completely. Homework: P. 228 1-21 odd.

Purpose: To factor polynomials completely. Homework: P. 228 1-21 odd.

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