1. Polynomial Functions Chapter 6

2. Polynomial Functions Variable – a symbol (letter) that represents a quantity that can vary Constant – a symbol that represents a specific number/doesn’t vary Term – constant, variable or a product of a constant and one or more variables

3. More Definitions Monomial – constant, variable or a product of a constant and one or more variables raised to counting number powers (1,2,3…) Polynomial – monomial or sum of monomials

4. Polynomial Properties Polynomials are written in descending order of the powers of the variables If there are multiple variables –v–variables are put in alphabetical order –t–terms are written in descending order according to the power of the variable that comes first in the alphabet

5. Degree One variable –e–exponent of the variable –t–two or more variables

6. Degree (cont) term with no variables degree of the polynomial –l–largest degree of any nonzero term Note: it’s not necessarily the term on the left degree

7. Naming Polynomials 1 st degreeLinear polynomial 2 nd degreeQuadratic polynomial 3 rd degreeCubic polynomial

8. Coefficients –c–constant factor –l–leading coefficient coefficient of the largest degree -3 leading coefficient

9. Like Terms –constant terms or variable terms with the same variables raised to the same powers Like TermsUnlike Terms

10. Add coefficients of like terms

11. Adding/Subtracting Polynomials Note: When subtracting distribute (-) i.e. (-1) first

12. Polynomial Function –f–function expressed as f(x) = P –w–where P is a one variable polynomial Quadratic Function –f–function whose equation is in the form –w–where a ≠ 0 –N–Note: This is known as Standard Form

13. Evaluating Functions (Review)

14. Graphing Quadratic Functions parabola minimum point (lowest point) a > 0, opens up maximum point (highest point) a < 0, opens down vertex (highest or lowest point) axis of symmetry (vertical line passing through the vertex)

15. Cubic Functions an equation that can be written in the form where a ≠ 0.

16. Graphing Cubic Functions

17. Sum/Difference of Functions Sum Difference

20. Modeling Situations with Sum/Difference of Functions YearF(s)M(s) 1993 (3)2518 1996 (6)2821 1998 (8)3022 2000 (10)3127 2002 (12)3228 2005 (15)3630

21. Find (W+M)(s) Find (W+M)(25) What does this mean? –T–There will be approximately 83 students in 2015

22. Find (W – M)(s) Find (W – M)(30) What does this mean? –There will be approximately 3 more woman students than male students in 2020

23. Find (W-M)(60) Is a negative number acceptable answer? –Y–Yes Why? –S–Since we are calculating how many more women students than male, a negative number represents more male than female students

24. 6.2 Multiplying Polynomial Expressions and Functions

25. Classifying Polynomials MonomialBinomialTrinomial One nonzero term Two nonzero terms Three nonzero terms

26. Product of Polynomials

27. Product of Polynomials (cont)

29. Squaring a Binomial Square of Sums Square of differences

30. Examples

31. Product of Binomial Conjugates

32. Evaluating Functions

33. Product of Functions

34. 6.3 Factoring Trinomials

35. Factoring If then (x+2) and (x+3) are factors of the polynomial Factor a polynomial – to write it as a product of polynomials Multiplying Factoring

36. For the polynomial: Look for two integers p and q whose product is c (pq = c) and whose sum is b (p + q = b). If integers exist then:

37. Factoring Example Product = 20Sum = 9 Factors of the polynomial are: Two +’s terms means both factors will have +’s

38. Factoring Example Product = 20Sum = -9 Factors of the polynomial are: Middle term (-), Last term (+) then both factors will be (-)’s

39. Factoring Example Product = 20Sum = -1 Factors of the polynomial are: If both terms (-)’s, then the larger number of the factors will be negative

40. Factoring Example Product = 20Sum = -1 Factors of the polynomial are: If both terms (-)’s, then the larger number of the factors will be negative

41. Factoring Example Product = 20Sum = 1 Factors of the polynomial are: If middle term + and last term (-), then the larger number of the factors will be positive

42. Factoring Two Variables Product = 20Sum = 9 Factors of the polynomial are:

43. Prime Polynomial Polynomial that can’t be factored Product = 15Sum = 3 No solutions, so it’s prime

44. Factoring out GFC Greatest Common Factor (GFC) – is the monomial with the largest coefficient and the highest degree that is a factor of all terms

45. Completely Factoring

46. 6.4 Factoring Polynomials

47. Factoring by Grouping

48. Factoring Trinomials by Grouping In the form 1.Find the pairs of numbers whose product is ac. 2.Determine which pair has sum b. Call the pair m and n. 3.Write the bx as mx + nx. 4.Factor by grouping. (ac method)

49. Example

51. 6.5 Factoring Special Binomials

52. Difference of Two Squares

53. Prime Binomials Binomial in the form is prime. Warning:

54. Sum/Difference of Cubes Sum of two cubes Difference of two cubes

55. Examples

57. 6.6 Using Factoring to Solve Polynomial Equations

58. Zero Factor Property Let A and B be real numbers, –If AB = 0, then A = 0 or B = 0.

59. Solving Quadratic Equation

60. x-intercepts of Quadratic Equations

61. More Examples

62. x-intercepts of Quadratic Equations Two SolutionsOne Solution No Solution

63. Solving Quadratic Equation

64. Solving Cubic Equations for x-intercepts

65. Quadratic Graphs Three SolutionsOne Solution Two Solutions

66. Area Problem A gardener has a rectangular garden with width of 8 feet and a length of 12 feet. To form a border of uniform width, she plans to place mulch around the border. If she has just enough sod to cover 44 square feet, determine the width of the border. 8 12 x x x x x + 12 + x = 2x+ 12 x + 8 + x = 2x+ 8