2. Chapter 11 Polynomials

3. 11-1 Add & Subtract Polynomials

4. Monomial A constant, a variable, or a product of a constant and one or more variables -7 5u (1/3)m 2 -s 2 t 3

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

6. Trinomial l A polynomial that has three terms 2x 2 – 3x + 1 14 + 32z – 3x mn – m 2 + n 2

7. Polynomial Expressions with several terms that follow patterns. 4x 3 + 3x 2 + 15x + 2 3b 2 – 2b + 4

8. Coefficient l The constant (or numerical) factor in a monomial l 3m 2 coefficient = 3 l u coefficient = 1 l -s 2 t 3 coefficient = -1

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

10. Examples l x 2 + (-4)x + 5 l x 2 – 4x + 5 l What are the terms? l x 2, -4x, and 5

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

12. Are they Simplified? l 2x 2 – 5 + 4x + x 2 l 3x + 4x – 5 l 4x 2 – x + 3x 2 – 5 + x 2

13. 11-2 Multiply by a Monomial

14. Examples l (5a)(-3b) l 3v 2 (v 2 + v + 1) l 12a(a 2 + 3ab 2 – 3b 3 – 10)

15. 11-3 Divide and Find Factors

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

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

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

19. Find the GCF of 12 and 36 12 = 36 = GCF =

20. Find the GCF of 14,49 and 56 14 = 49 = 56 = GCF =

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

22. 21x 2 – 35y 2 = Factoring Polynomials

23. 13e – 39ef = Factoring Polynomials

24. 5m + 35 5 = 5(m+ 7)÷5 = m + 7 Dividing Polynomials by Monomials

25. 7x + 14 7 = 7x + 14 7 7 = x + 2 Dividing Polynomials by Monomials

26. 6a + 8b 2 = 2(a +4b) ÷ 2 = a + 2b Dividing Polynomials by Monomials

27. 2x + 6x 2 2x Dividing Polynomials by Monomials

28. 11-4 Multiply Two Binomials

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

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

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

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

33. Example: (x + 6)(x +4) l F – x ▪ x l O – x ▪ 4 l I – 6 ▪ x l L – 6 ▪ 4

34. 11-5 Find Binomial Factors in a Polynomial

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

36. 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

37. Example 4x 3 + 4x 2 y 2 + xy + y 3 Group (4x 3 + 4x 2 y 2 ) and factor Group (xy + y 3 ) and factor 4x 2 (x +y 2 ) + y(x + y 2 ) Answer: (x +y 2 ) (4x 2 + y)

38. Example 2x 3 - 2x 2 y - 3xy 2 + 3y 3 + xz 2 – yz 2 Group (2x 3 - 2x 2 y 2 ) and factor Group (- 3xy 2 + 3y 3 ) and factor Group (xz 2 – yz 2 ) and factor Answer:

39. 11-6 Special Factoring Patterns

40. 11-6 Difference of Squares (a + b)(a – b)= a 2 - b 2 (x + 5) (x – 5) = x 2 - 25

41. 11-6 Squares of Binomials (a + b) 2 = a 2 + 2ab + b 2 (a - b) 2 = a 2 - 2ab + b 2 Also known as Perfect square trinomials

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

43. 11-7 Factor Trinomials

44. Factoring Pattern for x 2 + bx + c, c positive x 2 + 8x + 15 = l Find factors of 15 that add to 8 l Replace 8 with the added factors l Factor by Grouping

45. Example y 2 + 14y + 40 =

46. Example y 2 – 11y + 18 =

47. Factoring Pattern for x 2 + bx + c, c negative x 2 - x - 20 =

48. Example y 2 + 6y - 40 =

49. Example y 2 – 7y - 18 =

50. 11-9 More on Factoring Trinomials

51. 11-9 Factoring Pattern for ax 2 + bx + c Multiply a(c) = ac List the factors of ac Identify the factors that add to b Rewrite problem and factor by grouping

52. Example 2x 2 + 7x – 9 List factors: (-2)(9) = -18 Factors: (-2)(9) add to 7 (2x 2 -2x) + (9x – 9) 2x(x -1) + 9(x – 1) (x-1)(2x +9)

53. Example 14x 2 - 17x + 5 List factors: (14)(5) = 70 Factors: (-7)(-10) add to -17 14x 2 -7x – 10x + 5 (14x 2 – 7x) + (-10x +5) 7x(2x-1)- 5(2x -1) (7x -5)(2x – 1)

54. Example 3x 2 - 11x - 4 List factors: (-12)(1) = -12 Factors: (-12)(1) add to -11 3x 2 -12x + 1x - 4 (3x 2 – 12x) + (1x -4) 3x(x-4) + 1(1x -4) (x -4)(3x + 1)

55. END