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Chapter 5

Chapter 5. Factoring Polynomials. 5-1 Factoring Integers. Factors - integers that are multiplied together to produce a product. 4 x5 = 20. 2,3,5,7,11,13,17,19,23,29. Prime number - is an integer greater than 1 that has no positive integral factor other than itself and 1.

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Chapter 5

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  1. Chapter 5 Factoring Polynomials

  2. 5-1 Factoring Integers Factors - integers that are multiplied together to produce a product. 4 x5 = 20

  3. 2,3,5,7,11,13,17,19,23,29 Prime number - is an integer greater than 1 that has no positive integral factor other than itself and 1.

  4. PRIME FACTORIZATION Prime factorization of 36 36 = 2 x 18 = 2 x 2 x 9 = 2 x 2 x 3 x 3 = 22 x 32

  5. GREATEST COMMON FACTOR The greatest integer that is a factor of all the given integers.

  6. GREATEST COMMON FACTOR Find the GCF of 25 and 100 25 = 5 x 5 100 = 2 x 2 x 5 x 5 GCF = 5 x 5 = 25

  7. 5-2 Dividing Monomials

  8. Property of Quotients If a, b, c and d are real numbers, then ac =a • c bd b • d

  9. Simplifying Fractions If b, c and d are real numbers, then bc =c bd d

  10. Rule of Exponents for Division If a is a nonzero real number and m and n are positive integers, and m > n, then am= am-n an

  11. Rule of Exponents for Division If a is a nonzero real number and m and n are positive integers, and n > m; then am= 1 an an-m

  12. Rule of Exponents for Division If a is a nonzero real number and m and n are positive integers, and m = n; then am= 1 an

  13. GREATEST COMMON FACTOR The greatest common factor of two or more monomials is the common factor with the greatest coefficient and the greatest degree in each variable.

  14. GREATEST COMMON FACTOR Find the GCF of 25x4y and 50x2y5 GCF = 25x2y

  15. 5-3 Monomial Factors of Polynomials

  16. Dividing a Polynomial by a Monomial • Divide each term of the polynomial by the monomial and add the results.

  17. Dividing Polynomials by Monomials • 5m + 35 = m + 7 5 • 7x2 + 14x = x + 2 7x

  18. Factoring a Polynomial To factor: • Find the GCF • Divide each term by the GCF • Write the product

  19. Examples • 5x2 + 10x • 4x5 – 6x3 + 14x • 8a2bc2 – 12ab2c2

  20. 5-4 Multiplying Binomials Mentally When multiplying two binomials both terms of each binomial must be multiplied by the other two terms

  21. Binomial • A polynomial that has two terms 2x + 3 4x – 3y 3xy – 14 613 + 39z

  22. Trinomial • A polynomial that has three terms 2x2 – 3x + 1 14 + 32z – 3x mn – m2 + n2

  23. Multiplying binomials • Using the F.O.I.L method helps you remember the steps when multiplying

  24. F.O.I.L. Method • F – multiply First terms • O – multiply Outer terms • I – multiply Inner terms • L – multiply Last terms • Add all terms to get product

  25. Example:(2a – b)(3a + 5b) • F – 2a · 3a • O – 2a · 5b • I – (-b) ▪ 3a • L - (-b) ▪ 5b • 6a2 + 10ab – 3ab – 5b2 • 6a2 + 7ab – 5b2

  26. Example:(x + 6)(x +4) • F – x ▪ x • O – x ▪ 4 • I – 6 ▪ x • L – 6 ▪ 4 • x2 + 4x + 6x + 24 • x2 + 10x + 24

  27. Section 5-5 Difference of Two Squares

  28. Multiplying (x + 3) (x - 3) = ? (y - 2)(y + 2) = ? (s + 6)(s – 6) = ?

  29. Factoring Pattern a2 – b2 =(a –b) (a + b)

  30. FACTOR x2 - 49 = ? 16 – y2 = ? 81t2 – 25x6 = ?

  31. 5-6 Squares of Binomials

  32. Examples - Multiply (x + 3)2 = ? (y - 2)2 = ? (s + 6)2 = ?

  33. Factoring Patterns (a + b)2 = a2 + 2ab + b2 (a - b)2 = a2 - 2ab + b2 • Also known as Perfect square trinomials

  34. Examples – Factor • 4x2 + 20x + 25 • 64u2 + 72uv + 81v2 • 9m2 – 12m + 4 • 25y2 + 5y + 1

  35. 5-7 Factoring Pattern for x2 + bx + c, c positive

  36. Example x2 + 8x + 15 Middle term is the sum of 3 and 5 Last term is the product of 3 and 5

  37. Example y2 + 14y + 40 Middle term is the sum of 10 and 4 Last term is the product of 10 and 4

  38. Example y2– 11y + 18 Middle term is the sum of -2 and -9 Last term is the product of -2and -9

  39. Factor • m2 – 3m + 5 • k2 + 9k + 20 • y2 – 9y + 8

  40. 5-8 Factoring Pattern for x2 + bx + c, c negative

  41. x2 - x - 20 Middle term is the sum of 4 and -5 Last term is the product of 4 and -5

  42. Example y2 + 6y -40 Middle term is the sum of 10 and -4 Last term is the product of 10 and -4

  43. Example y2– 7y - 18 Middle term is the sum of 2 and -9 Last term is the product of 2 and -9

  44. Factor • x2 – 4kx – 12k2 • p2 – 32p – 33 • a2 + 3ab – 18b2

  45. 5-9 Factoring Pattern for ax2 + bx + c • List the factors of ax2 • List the factors of c • Test the possibilities to see which produces the correct middle term

  46. Examples • 2x2 + 7x – 9 • 14x2 - 17x + 5 • 10 + 11x – 6x2 • 5a2 – ab – 22b2

  47. 5 -10 Factor by Grouping • Factor each polynomial by grouping terms that have a common factor • Then factor out the common factor and write the polynomial as a product of two factors

  48. Examples • xy – xz – 3y + 3z • 3xy – 4 – 6x + 2y • xy + 3y + 2x + 6 • ab – 2b + ac – 2c • 9p2 – t2 – 4ts – 4s2

  49. 5 -11 Using Several Methods of Factoring • A polynomial is factored completely when it is expressed as the product of a monomial and one or more prime polynomials.

  50. Guidelines for Factoring Completely • Factor out the greatest monomial factor first • Factor the remaining polynomial

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