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Factoring (4.2):

Factoring (4.2):. Follow these basic steps …. Step 1:. Factor out the GCF. Step 2:. 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.

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Factoring (4.2):

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  1. Factoring (4.2): Follow these basic steps …

  2. Step 1: Factor out the GCF.

  3. Step 2: Count how many terms and try the following tactics. Then, go to step 3. • 2 terms -- • difference of 2 squares: a2 – b2 = (a + b)(a – b) Example: Factor 64x4 – 9y2  a = 8x2 and b = 3y = (8x2 + 3y)(8x2 – 3y)

  4. 2 terms -- SOMPS • difference of 2 cubes: a3 – b3 = (a - b)(a2 + ab + b2) SOMPS Example: Factor 8x3 – y3 Square first term Opposite sign Multiply Plus Square Second term = (2x - y)( 4x2 + 2xy + y2)

  5. 2 terms -- SOMPS • sum of 2 cubes: a3 + b3 = (a + b)(a2 - ab + b2) ** Notice that SOMPS still works here. Example: Factor 64s3 + t6 = (4s +t2)( 16s2 - 4st2 + t4) NOTE: a2 + b2 is NOT factorable.

  6. 3 terms -- • Set up 2 ( )’s • use factors of 1st term and last term until you get a pair that works with middle terms (guess and check) Signs must be different Example: Factor x2 - 7x - 18 1,18 2,9 3,6 1, 1 = (1x + 2)(1x – 9) Check middle terms 2x -9x -7x Matches middle term of original. Yea! 

  7. Signs must be the same Example: Factor 6x2 + 17x + 12 1,12 2,6 3,4 1, 6 2,3 = (3x + 4)(2x + 3) 8x 9x 17x ** This can be exhausting (trying to pick the factors that work)! Try alternate method

  8. 4 or more terms -- Try grouping Intro: • Factor xz – xy = x(z – y) (x + 2)(z – y) • Factor (x + 2)z – (x + 2)y = Example: Factor 2x + x2 – 6y – 3xy S1: group the terms – I pick the 1st and the 3rd terms; I reorder. = 2x – 6y + x2 – 3xy S2: Factor out the GCF from each pair = 2 + x (x – 3y) (x – 3y) S3: Since (x – 3y) is the same in both terms – factor it out. = (x – 3y)(2 + x)

  9. Step 3: Repeat steps until all factors are prime; i.e., they can’t be factored anymore.

  10. Example: Factor 4x6– 64x2 Step 1 – Factor out GCF = 4x2(x4 – 16) Step 2 – Count how many terms • 2 terms – it’s the difference of 2 squares = 4x2(x2 + 4)(x2 – 4) Step 3 – Keep repeating until all factors are prime = 4x2(x2 + 4)(x+ 2)(x - 2)

  11. Alternate Method to factoring trinomials … • this will ALWAYS work with a factorable trinomial Example: Factor 6x2 – x - 12 ax2 + 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

  12. Alternate Method to factoring trinomials … • this will ALWAYS work with a factorable trinomial Example: Factor 6x2 – x - 12  a = 1, b = -1, c = -12 STEP 4: Replace bx (in original) with factors. = 6x2 – 9x + 8x - 12 STEP 5: Group 1st two terms and last 2 terms. = 6x2 – 9x + 8x - 12 STEP 6: Factor. = 3x + 4 (2x – 3) (2x – 3) = (2x – 3)(3x + 4) Back to Notes

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