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Section P.4 Factoring

Learn effective strategies for factoring polynomials, including finding the greatest common factor, identifying difference of squares and perfect square trinomials, using inspection and grouping methods, and further factoring if possible.

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Section P.4 Factoring

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  1. Section P.4Factoring

  2. Strategy for Factoring a Polynomial • GCF • Two terms – check for the difference of squares • Three terms – check for perfect square trinomial • Inspection • m&n Grouping • More than three terms – Grouping • Can you factor further?

  3. Step 1 GCF • If the polynomial has a greatest common factor other than 1, then factor out the greatest common factor. 5x2 + 10x = ? 5x(x + 2)

  4. 14x + 21 = 9x – 12y = 2x2 + 6x + 4 = 5ab2 + 10a2b2 + 15a2b = 7(2x + 3) 3(3x – 4y) 2(x2 + 3x + 2) 5ab(b + 2ab + 3a) Step 1 GCF

  5. 5a + 7a = a(5 + 7) = a(12) = 12a y(x + 2) + 6(x + 2) =(x + 2)(y + 6) Here (x + 2) was the GCF Step 1 GCF

  6. Step 2 Difference of Squares (count the number of terms) • Count the number of terms. • If the polynomial has two terms (it is a binomial), then see if it is the difference of two squares. a2 – b2 = (a + b)(a – b) x2 – 9 = (x + 3)(x – 3) • Remember the sum of squares will not factor. a2 + b2

  7. Using FOIL we find the product of two binomials.

  8. Rewrite the polynomial as the product of a sum and a difference.

  9. Conditions for Difference of Squares • Must be a binomial with subtraction. • First term must be a perfect square. (x)(x) = x2 • Second term must be a perfect square (5)(5) = 25

  10. Recognizing the Difference of Squares • Must be a binomial with subtraction. • First term must be a perfect square (p)(p) = p2 • Second term must be a perfect square (10)(10) = 100

  11. Recognizing the Difference of Squares • Must be a binomial with subtraction. • First term must be a perfect square (3m)(3m) = 9m2 • Second term must be a perfect square (7)(7) = 49

  12. 4x2 – 25 = 2x2 – 8 = b2 + 100 = y4 – 16 = (2x + 5)(2x – 5) 2(x2 – 4) = Sum of Squares. Will not factor. (y2 + 4)(y2 – 4) = (y2 + 4)(y + 2)(y – 2) Step 2 Difference of Squares 2(x + 2)(x – 2)

  13. Step 3 Trinomials • Count the number of terms. • If there are three terms • Check to see if it is a perfect square trinomial. • Grouping method. (long) • Inspection (skill)

  14. Using FOIL we find the product of two binomials.

  15. Rewrite the perfect square trinomial as a binomial squared. So when you recognize this… …you can write this.

  16. Recognizing a Perfect Square Trinomial • First term must be a perfect square. (x)(x) = x2 • Last term must be a perfect square. (5)(5) = 25 • Middle term must be twice the product of the roots of the first and last term. (2)(5)(x) = 10x

  17. Recognizing a Perfect Square Trinomial • First term must be a perfect square. (m)(m) = m2 • Last term must be a perfect square. (4)(4) = 16 • Middle term must be twice the product of the roots of the first and last term. (2)(4)(m) = 8m

  18. Signs must match! Recognizing a Perfect Square Trinomial • First term must be a perfect square. (p)(p) = p2 • Last term must be a perfect square. (9)(9) = 81 • Middle term must be twice the product of the roots of the first and last term. (2)(-9)(p) = -18p

  19. Recognizing a Perfect Square Trinomial Not a perfect square trinomial. • First term must be a perfect square. (6p)(6p) = 36p2 • Last term must be a perfect square. (5)(5) = 25 • Middle term must be twice the product of the roots of the first and last term. (2)(5)(6p) = 60p ≠ 30p

  20. Using FOIL we find the product of two binomials.

  21. The two binomials represent the trinomial in factored form. Our job is to rewrite trinomials in factored form.

  22. This means that to find the correct factorization we must find two numbers m and n with a sum of 10 and a product of 24. Start with the trinomial and pretend that you have a factorization.

  23. Factoring a Trinomial by Grouping First list the factors of 24. Rewrite with four terms. Now add the factors. 25 1 24 2 12 14 3 8 11 10 4 6 Notice that 4 and 6 sum to the middle term.

  24. Factoring a Trinomial by Grouping First list the factors of 24. Rewrite with four terms. Now add the factors. 25 1 24 2 12 14 3 8 11 10 4 6 Notice that 2 and 12 sum to the middle term.

  25. Step 3 Inspection • Guess at the factorization until you get it right. • Check with multiplication.

  26. Step 4 Grouping • If the polynomial has more than three terms, try to factor by grouping.

  27. Step 5 • As a final check see if any of the factors you have written can be factored further. • If you have overlooked a common factor you can catch it here.

  28. Count • How long is the edge? • How many squares in the face? • How many blocks? 1 1 1

  29. Count 2 4 8

  30. Count 2 4 8 3 9 27

  31. Count n n2 n3 2 4 8 3 9 27 4 16 64 5 25 125

  32. Memorize the First 10 Perfect Cubes

  33. Multiply a Binomial by a Trinomial The Sum of Cubes

  34. Difference of Cubes

  35. Compare the Formulas The Sum of Cubes The Difference of Cubes They are just alike except for where they are different.

  36. Using the Difference of Cubes x3 - 8 Recall 23 = 8 = (x - 2)(x2 + 2x + 4)

  37. Using the Sum of Cubes y3 + 27 Recall 33 = 27 = (y + 3)(y2 – 3y + 9)

  38. Getting Ready for Class • What is the first step in factoring any polynomial? • If a polynomial has four terms, what method of factoring should you try? • If a polynomial has two terms, what method of factoring should you try? • What is the last step in factoring a polynomial?

  39. Homework Problem Set P.4 Pages 39 13- 33 odd49-55 odd 61-69 odd 79, 81

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