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5.4 Factoring Polynomials

5.4 Factoring Polynomials. Factoring GCF Group factoring Factoring Trinomials Special Cases. Problem 1. The first thing to do when factoring. Find the greatest common factor (GCF), if there is one. This is a number that can divide into all the terms of the polynomial. .

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5.4 Factoring Polynomials

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  1. 5.4 Factoring Polynomials Factoring GCF Group factoring Factoring Trinomials Special Cases

  2. Problem 1 The first thing to do when factoring Find the greatest common factor (GCF), if there is one. This is a number that can divide into all the terms of the polynomial. What is the GCF for this problem? 8 Now divide each term by 8 and what will be left?

  3. Problem 2 What is the GCF for this problem? (DO NOT COMBINE LIKE TERMS ON THIS ONE) NOTE:Just because you had a GCFdoes NOT mean you are done factoring. 2 Now divide each term by 2 and what will be left? • When you have 4 terms it is called “Factor by Grouping.” You will split the 4 terms in half and follow the GCF method again on BOTH sides of the equation. This method will be used quite often. At this point, you can leave the GCF of 2 from above alone until the FINAL answer. What is the GCF on each side of the line? Left side: 2x Right side: -5

  4. Now divide each side by that GCF. What do you notice about the parenthesis? The parenthesis are now the GCF. So divide that out. And the 2 GCF’s on the outside of the parenthesis are the ones that are left. ( 2x - 5 ) Final Factored Form: 2 DONT FORGET OUR FIRST GCF 2

  5. Problem 3 Try this one! Group factoring What is the GCF on each side of the line? Left side: Right side: - 2 x2 (x + 5) – 2(x + 5) Final Factored Form:

  6. Problem 4 Factor x3 + 4x2 + 2x + 8 x2 ( x + 4) + 2(x + 4) (x + 4)(x2 + 2) Final Factored Form:

  7. Problem 5 Think about how you would factor 3y2– 2y – 5? I would turn it into a “factor by grouping” problem First, multiply the ends together, 3 times – 5 is -15. Second, what are the factors of -15? And which of those factors multiply to be -15 and add to be – 2 - 5 and 3 Therefore, I need to break/split the middle term into -5y and +3y 3y2- 5y + 3y – 5

  8. 3y2- 5y + 3y – 5 y(3y – 5) + 1(3y – 5) ((3y – 5)(y + 1) Final Factored Form:

  9. You try one! 6x2 - 7x - 3 Final Factored Form:

  10. Find the numbers that multiply to the given number 24 24 The factors of 24 are 1 24 2 12 3 8 4 6 Now consider this trinomial (3 Terms): x2 + 11x + 24 Multiply the end together, 1 times 24. Notice the first term is a 1. There are only one set of numbers that are factors of 24 and add to 11 Which ones are they?

  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. FINAL FACTORED FORM When the first term is a 1, the numbers go directly into the parenthesis! No need to break/split the middle term.

  12. You Try One!

  13. Special Case: Difference of Two Squares Factor x2 – 25 Here you find what factors multiply to be -25 and adds to be 0 (the middle term is not there). Therefore, the number must be negative and positive. ( x + __)(x - __)

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

  15. Another Special case: Perfect Square Trinomialsx2 + 2xy + y2 = (x + y)2x2 - 2xy + y2 = (x - y)2 4x2– 20xy + 25y2 Notice the first and last terms are perfect squares (2x)2 – 2(2x)(5y) + (5y)2 (2x – 5y)2

  16. Sum and Difference of Two Cubes Factor: x3 +125 For the Binomial Factor Ask: • What term to the third power equals x 3 ? • What is the given sign in the problem? • What number to the third power equals 125? x Plus (+) 5 The Binomial factor is (x+ 5) Square the front/first term of the binomial factor x2 Opposite sign of the binomial factor - (minus) First term times the second term of the binomial factor 5x Always positive Plus (+) 52 Square the back/second term of the binomial factor FINAL FACTORED FORM

  17. Sum and Difference of Two Cubes Factor: 8y3- 125 For the Binomial Factor Ask: • What term to the third power equals 8y3 ? • What is the given sign in the problem? • What number to the third power equals 125? 2y - (minus) 5 The Binomial factor is (2y - 5) Square the front/first term of the binomial factor 4y2 Opposite sign of the binomial factor + (plus) First term times the second term of the binomial factor 10y Always positive (plus sign) + (plus) Square the back/second term of the binomial factor 25 FINAL FACTORED FORM

  18. You Try Some! x3+ 343y3= ** 8k3– 64c3= Be careful with this one!!

  19. Homework Due Monday Page 338-339 #22-33, 37-46, 48-53 Must show your work

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