1. Bell Work:

2. UNIT 4 Polynomial and Rational Functions

3. LESSON 4.1 Polynomial Functions with Degree Greater Than 2

4. Degree of a Polynomial The degree of a polynomial is the highest order exponent that the function has. The functions we will be looking at in this lesson have a degree that is greater than two, which means the functions will not be linear or quadratic.

5. Intermediate Value Theorem: We can prove that there is a zero for a polynomial function by using this theorem. If f(a) > 0 and f(b) < 0, then there must be a point between a and b that crosses the x-axis, which means there is a zero between a and b. Ex: If f(x) = 2x³ - 2, show that there is a zero for the function between -2 and 2.

6. Sketching the Graph

7. Bell Work:

9. Class Work: Pages 269-271 #’s 33, 35, 39, 41

10. Homework: Pages 269-271 #’s 5-9 odds, 17-27 odds, 34, 36, 43

11. Bell Work:

12. LESSON 4.2 Dividing Polynomials

13. LESSON ESSENTIAL QUESTION What are the different properties of division pertaining to polynomials and how are the used?

14. Long Division of Polynomials

15. Bell Work:

16. More Examples:

17. Synthetic Division of Polynomials

18. Remainder Theorem:

19. Factor Theorem:

20. Homework: Pages 278-279 #’s 1, 2, 9, 11, 13, 15, 21, 23, 25, 29, 31, 35, 37 Small Quiz Tomorrow: Finding Zeros, Sketching Graphs, Intervals Long Division/Synthetic Division

21. Bell Work:

22. Class Work/Homework: Pages 278-279 #’s 10, 12, 14, 16, 22, 24, 26, 28, 30, 36

23. Bell Work:

24. LESSON 4.3 Zeros of Polynomials

25. LESSON ESSENTIAL QUESTION How do we find the total number of zeros for a function, and how do we determine the number of real/imaginary, positive/negative, rational/irrational zeros?

26. Multiplicity of Zeros

27. Finding the Polynomial We can find a polynomial f(x) if we know its zeros and one other point on the function. Ex: Find f(x) if f(-4) = -504 and it has zeros: -1 with multiplicity of 2, 3, 4 Ex: Find f(x) if f(5) = 68 and it has zeros: ±3i, 0, 3

28. Finding Zeros

29. Homework: Pages 291-293 #’s 1 – 25 odds

30. Bell Work:

31. Classwork/Homework: Pages 291-293 #’s 2, 6, 10, 12, 14, 16, 18, 22, 24 This assignment will be collected tomorrow at the beginning of class!!!

32. Bell Work: Warning, the lesson you are about to learn is off the heezy fo sheezy…viewer discretion may be advised.

33. LESSON 4.4 Complex and Rational Zeros of Polynomials

34. Lesson Essential Question: How do we determine the rational and complex solutions to a polynomial function?

35. Find the function knowing the zeros… Example: Find a polynomial function f(x) of degree 4 that has zeros of (2 + i) and (-3i) that has a leading coefficient of 1. Example: Find a polynomial function f(x) of degree 5 that has zeros of (2), (5i) and (2 + 3i) that has a leading coefficient of 1.

36. Dear 4 th Period Pre-Calculus Class, I would like to apologize for a mistake that I made yesterday. I incorrectly typed the notes into the PowerPoint slide which caused confusion at the end of class. I hope that some day you all can forgive me for this horrific and tragic error that I have made. I know it is hard to understand because you all see me as a superhuman math machine, but like all superheroes, I have flaws. Am I implying I am a superhero? I guess so. Anyway, back to the point. I sincerely apologize and beg for your forgiveness. Sincerely, Mr. Kelsey (aka Math Man aka The Fabulous Factorer)

37. Determining Possible Zeros…

38. Example:

40. Homework: Pages 301 – 302 #’s 1, 5, 9, 15, 17, 19, 23

41. Bell Work:

42. Classwork/Homework: Pages 301 – 302 #’s 2, 6, 10, 18, 20, 24 This will be collected tomorrow!!!

43. Bell Work:

44. Warning: The lesson you are about to witness is off the hook, which can be extremely dangerous and not appropriate for all audiences. Viewers discretion may be advised.

45. LESSON 4.5 Behaviors of Rational Functions

46. Lesson Essential Question: How do we determine the horizontal, vertical, and oblique asymptotes for a rational function, and the behavior about those asymptotes?

47. Let’s try some examples…

48. Bell Work:

49. Holes!!!!! If a factor from the numerator and denominator will cancel out, then you do not have a vertical asymptote. Instead, you have a hole! This is a value that cannot exist because of the factor in the denominator, but would have a value based upon the simplified form of the function.

50. Vertical Asymptotes:

51. Horizontal Asymptotes:

52. Bell Work:

53. Oblique Asymptotes:

54. Steps for Graphing Rational Functions:

55. Examples:

56. Homework: Pages 318 – 319 #’s 7, 9, 15, 17, 21, 25, 29, 33

57. Quiz on Rational Functions:

58. Bell Work:

59. Examples:

60. Unit 4 Test Upcoming! The Test will be on: Sketching Polynomial Functions Using its Zeros Find Intervals in which f(x) is positive or negative Division of Polynomials (Long and Synthetic) Remainder and Factor Theorem Finding Zeros of Polynomials (Real, Complex, Irrational) Find the Polynomial based upon its zeros and an additional point Sketching graphs of Rational Functions

61. Class Work: Pages 321-322 #’s 3-6, 10-13, 15-20, 24 – 27, 29, 31, 33, 35