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Chapter 11

Chapter 11. Polynomials. 11-1. Add & Subtract Polynomials. Monomial. A constant, a variable, or a product of a constant and one or more variables -7 5 u (1/3)m 2 -s 2 t 3. Binomial. A polynomial that has two terms 2x + 3 4x – 3y 3xy – 14 613 + 39z.

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Chapter 11

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  1. Chapter 11 Polynomials

  2. 11-1 Add & Subtract Polynomials

  3. Monomial A constant, a variable, or a product of a constant and one or more variables -7 5u (1/3)m2 -s2t3

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

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

  6. Polynomial Expressions with several terms that follow patterns. 4x3 + 3x2 + 15x + 2 3b2 – 2b + 4

  7. Coefficient • The constant (or numerical) factor in a monomial • 3m2 coefficient = 3 • u coefficient = 1 • -s2t3 coefficient = -1

  8. Like Terms • Terms that are identical or that differ only in their coefficients • Are 2x and 2y similar? • Are -3x2 and 2x2 similar?

  9. Examples • x2 + (-4)x + 5 • x2 – 4x + 5 • What are the terms? • x2, -4x, and 5

  10. Simplified Polynomial • A polynomial in which no two terms are similar. • The terms are usually arranged in order of decreasing degree of one of the variables

  11. Are they Simplified? • 2x2 – 5 + 4x + x2 • 3x + 4x – 5 • 4x2 – x + 3x2 – 5 + x2

  12. 11-2 Multiply by a Monomial

  13. Examples • (5a)(-3b) • 3v2(v2 + v + 1) • 12(a2 + 3ab2 – 3b3 – 10)

  14. 11-3 Divide and Find Factors

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

  16. 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.

  17. 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

  18. GREATEST COMMON FACTOR Find the GCF of 12 and 36 12 = 36 = GCF =

  19. GREATEST COMMON FACTOR Find the GCF of 14,49 and 56 14 = 49 = 56 = GCF =

  20. Factoring Polynomials vw + wx = w(v + x)

  21. Factoring Polynomials 21x2 – 35y2 =

  22. Factoring Polynomials 13e – 39ef =

  23. Dividing Polynomials by Monomials 5m + 35 5 = 5(m+ 7)÷5 = m + 7

  24. Dividing Polynomials by Monomials 7x + 14 7 = 7x + 14 7 7 = x + 2

  25. Dividing Polynomials by Monomials 6a + 8b 2 = 2(a +4b) ÷ 2 = a + 2b

  26. Dividing Polynomials by Monomials 2x + 6x2 2x

  27. 11-4 Multiply Two Binomials

  28. Multiplying Binomials When multiplying two binomials both terms of each binomial must be multiplied by the other two terms

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

  30. 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

  31. Example:(2a – b)(3a + 5b) • F – 2a · 3a • O – 2a · 5b • I – (-b) ▪ 3a • L - (-b) ▪ 5b

  32. Example:(x + 6)(x +4) • F – x ▪ x • O – x ▪ 4 • I – 6 ▪ x • L – 6 ▪ 4

  33. 11-5 Find Binomial Factors in a Polynomial

  34. Procedure • Group the terms in the polynomial as pairs that share a common monomial factor • Extract the monomial factor from each pair

  35. Procedure • If the binomials that remain for each pair are identical, write this as a binomial factor of the whole expression • The monomials you extracted create a second polynomial. This is the paired factor for the original expression

  36. Example 4x3 + 4x2y2 + xy + y3 Group (4x3 + 4x2y2) and factor Group (xy + y3)and factor 4x2(x +y2) + y(x + y2) Answer: (x +y2) (4x2 + y)

  37. Example 2x3 - 2x2y - 3xy2 + 3y3+ xz2 – yz2 Group (2x3 - 2x2y2 ) and factor Group (- 3xy2 + 3y3)and factor Group (xz2 – yz2)and factor Answer:

  38. 11-6 Special Factoring Patterns

  39. 11-6 Difference of Squares (a + b)(a – b)= a2 - b2 (x + 5) (x – 5) = x2 - 25

  40. 11-6 Squares of Binomials (a + b)2 = a2 + 2ab + b2 (a - b)2 = a2 - 2ab + b2 • Also known as Perfect square trinomials

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

  42. 11-7 Factor Trinomials

  43. Factoring Pattern for x2 + bx + c, c positive x2 + 8x + 15 = (x + 3) (x + 5) Middle term is the sum of 3 and 5 Last term is the product of 3 and 5

  44. Example y2 + 14y + 40 = (y + 10) (y + 4) Middle term is the sum of 10 and 4 Last term is the product of 10 and 4

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

  46. Factoring Pattern for x2 + bx + c, c negative x2 - x - 20 = (x + 4) (x - 5) Middle term is the sum of 4 and -5 Last term is the product of 4 and - 5

  47. Example y2 + 6y - 40 = (y + 10) (y - 4) Middle term is the sum of 10 and -4 Last term is the product of 10 and - 4

  48. Example y2 – 7y - 18 =(y + 2) (y - 9) Middle term is the sum of 2 and -9 Last term is the product of 2 and -9

  49. 11-9 More on Factoring Trinomials

  50. 11-9 Factoring Pattern for ax2 + bx + c • Multiply a(c) = ac • List the factors of ac • Identify the factors that add to b • Rewrite problem and factor by grouping

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