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Comprehensive Guide to Factoring Polynomials and Quadratics

This guide provides a step-by-step approach to factoring polynomials and quadratic equations. It covers essential techniques such as identifying whether the second term in a polynomial is positive or negative, utilizing the box method, and applying the difference of squares and factoring by grouping. Through practical examples, learn how to factor expressions like (x^2 + 5x + 6) and (x^2 - 9). Master the intricacies of polynomial expressions and enhance your algebra skills with this comprehensive resource.

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Comprehensive Guide to Factoring Polynomials and Quadratics

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  1. x2 36 2x2 -15 (x – 4) (x – 9) (2x– 5) (x + 3) -4x -5x -9x +6x X2 – 13x + 36 2X2+ x - 15

  2. 3x2 8 x2 -25 (x + 2) (3x + 4) (x – 5) (x + 5) 6x -5x 4x +5x 3X2 + 10x + 8 x2 - 25

  3. If the second is a plus, two of the first If the second is a plus, two of the first If the second is a plus, then you add to get the middle If the second is plus, two of the first Two of the first Second is a plus ( x2 +4x + 3) ( + ) ( + ) (x2 - 4x + 3 ) ( - ) ( - ) If the second is a minus, one of each If the second is a minus, one of each If the second is a minus, then you subtract to get the middle If the second is a minus, one of each. Second is a minus ( x2 +2x - 3) ( + ) ( - ) (x2 - 2x - 3 ) ( + ) ( - )

  4. If the second is a plus, two of the first.. you add to get the middle…. 2 of the first +’s 2nd is plus Factoring Polynomials X2 + 5x + 6 ( + )( + ) x 3 x 2 Add to get the 5 x, x 6, 1 6x+1x = 7x x, x 3,2 3x+2x = 5x

  5. If the second is a plus, two of the first.. you add to get the middle…. 2 of the first +’s 2nd is plus X2 + 9x + 20 ( + )( + ) x 5 x 4 Add to get the 9 x, x 10, 2 10x+2x = 12x x, x 5,4 5x+4x = 9x

  6. If the second is a plus, two of the first.. you add to get the middle…. 2 of the first +’s 2nd is plus X2 + 15x + 36 ( + )( + ) x 3 x 12 Add to get the 15 x, x 18, 2 2x+18x = 20x x, x 3,12 3x+12x = 15x

  7. If the second is a plus, two of the first.. you add to get the middle…. 2 of the first -’s 2nd is plus X2 - 9x + 20 ( - )( - ) x 5 x 4 Add to get the 9 x, x 10, 2 10x+2x = 12x x, x 5,4 5x+4x = 9x

  8. If the second is a plus, two of the first.. you add to get the middle…. 2 of the first -’s 2nd is plus X2 - 10x + 24 ( - )( - ) x 6 x 4 Add to get the 10 x, x 12, 2 12x+2x = 14x x, x 6,4 6x+4x = 10x

  9. If the second is a minus, one of each.. you subtract to get the middle +12x – 2x = 10x +2x – 12x = -10x 2nd is minus one of each X2 - 10x - 24 ( ) ( ) x 12 x 2 +, - - + Subtract to get the 10 x, x 6, 4 6x- 4x = 2x x, x 12,2 12x- 2x = 10x

  10. If the second is a minus, one of each.. you subtract to get the middle +10x – 2x = 8x +2x – 10x = -8x 2nd is minus one of each X2 + 8x - 20 ( ) ( ) x 10 x 2 +, - + - Subtract to get the 8 x, x 5, 4 5x- 4x = x x, x 10,2 10x- 2x = 8x

  11. If the second is a minus, one of each.. you subtract to get the middle +6x – 5x = x +5x –6x = -x 2nd is minus one of each X2 - x - 30 ( ) ( ) x 6 x 5 +, - - + Subtract to get the 1 x, x 15, 2 15x- 2x = 13x x, x 6,5 6x- 5x = x

  12. If the second is a minus, one of each.. you subtract to get the middle 2nd is minus one of each X2 - 36 + 0x ( ) ( ) x 6 x 6 +, - - + Subtract to get the 0 x, x 6, 6 6x- 6x = 0x A2 - B2 = (A – B)(A + B) x2 - 62 = (x – 6)(x + 6)

  13. If the second is a minus, one of each.. you subtract to get the middle 2nd is minus one of each X2 - 25 + 0x ( ) ( ) x 5 x 5 +, - - + Subtract to get the 0 x, x 5, 5 5x- 5x = 0x A2 - B2 = (A – B)(A + B) x2 - 52 = (x – 5)(x + 5)

  14. If the second is a plus, two of the first.. you subtract to get the middle Add to get 0!!!! X2 + 9 + 0x NOT FACTORABLE A2 - B2 = (A – B)(A + B)

  15. 1) x2 + 5x + 6 2) x2 - 5x + 6 3) x2 – 8x - 48 4) x2 + 7x - 30 5) x2 - 64 6) x2 - 9x + 14 7) x2 - 1 8) 4x2 - 2xy2 + 8x3 9) x2 + 16

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