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5.4 Factoring Polynomials

5.4 Factoring Polynomials. Group factoring Special Cases Simplify Quotients. The first thing to do when factoring. Find the greatest common factor (G.C.F), this is a number that can divide into all the terms of the polynomial. The first thing to do when factoring.

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5.4 Factoring Polynomials

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  1. 5.4 Factoring Polynomials Group factoring Special Cases Simplify Quotients

  2. The first thing to do when factoring Find the greatest common factor (G.C.F), this is a number that can divide into all the terms of the polynomial.

  3. The first thing to do when factoring Find the greatest common factor (G.C.F), this is a number that can divide into all the terms of the polynomial. Here it is 8

  4. Find the numbers that multiply to the given number The factors of 24 are 1 and 24 2 and 12 3 and 8 4 and 6 -1 and -24 -2 and -12 -3 and -8 -4 and -6

  5. There are only one set of numbers that are factors of 24 and add to 11 So in the trinomial x2 + 11x + 24 (x + ___)(x + ___) Since we add the like terms of the inner and outer parts of FOIL and multiply them to be the last number; the only numbers would work would be 3 and 8.

  6. Factor x2 _ 2x - 63

  7. Group factoring Here you will factor four terms, two at a time. When you find the common binomial that they have in common, factor it out of both parts. x3 + 5x2 – 2x – 10 x2 (x + 5) – 2(x + 5) (x + 5)(x2 -2)

  8. Group factoring Here you will factor four terms, two at a time. When you find the common binomial that they have in common, factor it out of both parts. x3 + 5x2 – 2x – 10 x2 (x + 5) – 2(x + 5) (x + 5)(x2 -2)

  9. Group factoring Here you will factor four terms, two at a time. When you find the common binomial that they have in common, factor it out of both parts. x3 + 5x2 – 2x – 10 x2 (x + 5) – 2(x + 5) (x + 5)(x2 -2)

  10. Factor x3 + 4x2 + 2x + 8 x2 ( x + 4) + 2(x + 4) (x + 4)(x2 + 2)

  11. Factor x3 + 4x2 + 2x + 8 x2 ( x + 4) + 2(x + 4) (x + 4)(x2 + 2)

  12. Factor x3 + 4x2 + 2x + 8 x2 ( x + 4) + 2(x + 4) (x + 4)(x2 + 2)

  13. How would you factor 3y2 – 2y -5 I would turn it into a group factoring problem

  14. How would you factor 3y2 – 2y -5 I would turn it into a group factoring problem Multiply the end together, 3 times – 5 is -15. What multiplies to be -15 and adds to – 2 - 5 and 3 So I break the middle term into -5y and +3y 3y2 - 5y + 3y – 5

  15. How would you factor 3y2 – 2y -5 3y2 - 5y + 3y – 5 y(3y – 5) + 1(3y – 5) (3y – 5)(y + 1)

  16. How would you factor 3y2 – 2y -5 3y2 - 5y + 3y – 5 y(3y – 5) + 1(3y – 5) (3y – 5)(y + 1)

  17. How would you factor 3y2 – 2y -5 3y2 - 5y + 3y – 5 y(3y – 5) + 1(3y – 5) (3y – 5)(y + 1)

  18. Homework Page 242 # 4 – 8 15 – 27 odd Must show work

  19. Special Case Factor x2 – 25 Here you find with multiply to be -25 and adds to be 0. The number must be negative and positive. ( x + __)(x - __)

  20. The rule is a2 – b2 = (a +b)(a – b) 16x2 – 81y4 (4x)2 – (9y2)2 = (4x + 9y2)(4x – 9y2)

  21. Another Special casex2 + 2xy + y2 = (x + y)2x2 - 2xy + y2 = (x - y)2 4x2 – 20xy + 25y2 (2x)2 – (2x)(5y) + (5y)2 (2x – 5y)2

  22. Another Special casex2 + 2xy + y2 = (x + y)2x2 - 2xy + y2 = (x - y)2 4x2 – 20xy + 25y2 (2x)2 – (2x)(5y) + (5y)2 (2x – 5y)2

  23. Another Special casex2 + 2xy + y2 = (x + y)2x2 - 2xy + y2 = (x - y)2 4x2 – 20xy + 25y2 (2x)2 – (2x)(5y) + (5y)2 (2x – 5y)2

  24. Sum of two CubesDifferent of Cubes a3 + b3 = (a + b)(a2 – ab + b2) x3 + 343y3 = (x + 7y)(x2 – x(7y) + (7y)2) x3 + 343y3 = (x + 7y)(x2 – 7xy + 49y2) a3 - b3 = (a - b)(a2 + ab + b2) 8k3 – 64c3 = (2k – 4c)((2k)2 + (2k)(4c) + (4c)2) 8k3 – 64c3 = (2k – 4c)(4k2 + 8ck + 16c2)

  25. Factor: x3y3 + 8 Sum of Cubes (xy)3 + (2)3 ((xy) + (2))((xy)2 – (xy)(2) + (2)2) (xy + 2)(x2y2 – 2xy +4)

  26. Factor: x3y3 + 8 Sum of Cubes (xy)3 + (2)3 ((xy) + (2))((xy)2 – (xy)(2) + (2)2) (xy + 2)(x2y2 – 2xy +4)

  27. Factor: x3y3 + 8 Sum of Cubes (xy)3 + (2)3 ((xy) + (2))((xy)2 – (xy)(2) + (2)2) (xy + 2)(x2y2 – 2xy +4)

  28. Factor: x3y3 + 8 Sum of Cubes (xy)3 + (2)3 ((xy) + (2))((xy)2 – (xy)(2) + (2)2) (xy + 2)(x2y2 – 2xy +4)

  29. Simplify Quotients Quotients are fractions with variables. Quotients can be reduced by factoring

  30. Simplify Quotients Quotients can be reduced by factoring Common factors can be crossed out

  31. Simplify Quotients Common factors can be crossed out

  32. Simplify Quotients Common factors can be crossed out Must stated that x cannot equal 1 or -8, why?

  33. Simplify Factor first

  34. Homework Page 242 – 243 # 16 – 28 even; 29, 33, 35, 39, 47, 51 Must show work

  35. Homework Page 242 – 243 # 30, 31, 32, 34, 37, 38, 44, 46, 48, 50 Must show work

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