1. Packet #7 Polynomial Functions
Math 160
Packet #7
Polynomial Functions

2. A polynomial function of degree 𝒏 is a function that can be written in the form: 𝑃 𝑥 = 𝑎 𝑛 𝑥 𝑛 + 𝑎 𝑛−1 𝑥 𝑛−1 +…+ 𝑎 1 𝑥+ 𝑎 0

3. Polynomial functions are continuous and smooth
Polynomial functions are continuous and smooth. That means no gaps, holes, cusps, or corners.

4. Polynomial functions are continuous and smooth
Polynomial functions are continuous and smooth. That means no gaps, holes, cusps, or corners.

5. Polynomial functions are continuous and smooth
Polynomial functions are continuous and smooth. That means no gaps, holes, cusps, or corners.

6. Polynomial functions are continuous and smooth
Polynomial functions are continuous and smooth. That means no gaps, holes, cusps, or corners.

7. Polynomial functions are continuous and smooth
Polynomial functions are continuous and smooth. That means no gaps, holes, cusps, or corners.

8. Polynomial functions are continuous and smooth
Polynomial functions are continuous and smooth. That means no gaps, holes, cusps, or corners.

9. Polynomial functions are continuous and smooth
Polynomial functions are continuous and smooth. That means no gaps, holes, cusps, or corners.

10. Polynomial functions are continuous and smooth
Polynomial functions are continuous and smooth. That means no gaps, holes, cusps, or corners.

11. The end behavior of a function means how the function behaves when 𝑥→∞ or 𝑥→−∞. For non-constant polynomial functions, the end behavior is either 𝑦→∞ or 𝑦→−∞. The highest degree term of a polynomial, called the ___________, determines its end behavior.

12. The end behavior of a function means how the function behaves when 𝑥→∞ or 𝑥→−∞. For non-constant polynomial functions, the end behavior is either 𝑦→∞ or 𝑦→−∞. The highest degree term of a polynomial, called the ___________, determines its end behavior.

13. The end behavior of a function means how the function behaves when 𝑥→∞ or 𝑥→−∞. For non-constant polynomial functions, the end behavior is either 𝑦→∞ or 𝑦→−∞. The highest degree term of a polynomial, called the ___________, determines its end behavior.

14. The end behavior of a function means how the function behaves when 𝑥→∞ or 𝑥→−∞. For non-constant polynomial functions, the end behavior is either 𝑦→∞ or 𝑦→−∞. The highest degree term of a polynomial, called the ___________, determines its end behavior.
leading term

15. Ex 1. Determine the end behavior of the polynomial 𝑃 𝑥 =−2 𝑥 4 +5 𝑥 3 +4𝑥−7.

16. Ex 1. Determine the end behavior of the polynomial 𝑃 𝑥 =−2 𝑥 4 +5 𝑥 3 +4𝑥−7.

17. Ex 2. Determine the end behavior of the polynomial 𝑃 𝑥 =3 𝑥 5 −2 𝑥 2 +𝑥−12212012.

18. Ex 2. Determine the end behavior of the polynomial 𝑃 𝑥 =3 𝑥 5 −2 𝑥 2 +𝑥−12212012.

19. Note: Zeros of a polynomial correspond with factors, and visually mean 𝑥-intercepts. ex: If 𝑓 𝑥 = 𝑥 2 −5𝑥+6, then since 𝑓 2 =0, we must have a factor of 𝑥−2. Also, there will be an 𝑥-intercept at 𝑥=2.

20. Note: Zeros of a polynomial correspond with factors, and visually mean 𝑥-intercepts. ex: If 𝑓 𝑥 = 𝑥 2 −5𝑥+6, then since 𝑓 2 =0, we must have a factor of 𝑥−2. Also, there will be an 𝑥-intercept at 𝑥=2.

21. Note: Zeros of a polynomial correspond with factors, and visually mean 𝑥-intercepts. ex: If 𝑓 𝑥 = 𝑥 2 −5𝑥+6, then since 𝑓 2 =0, we must have a factor of 𝑥−2. Also, there will be an 𝑥-intercept at 𝑥=2.

22. Graphing Polynomial Functions
Factor to find zeros and plot 𝒙-intercepts.
Plot test points (before smallest 𝑥-intercept, between 𝑥-intercepts, and after largest 𝑥-intercept).
Determine end behavior.
4. Graph.

23. Ex 3. Sketch the graph of 𝑓 𝑥 =− 𝑥 3 +2 𝑥 2 +3𝑥
Ex 3. Sketch the graph of 𝑓 𝑥 =− 𝑥 3 +2 𝑥 2 +3𝑥. Be sure to show intercepts, test points (before smallest 𝑥-intercept, between 𝑥-intercepts, and after largest 𝑥-intercept), and end behavior.

24. Ex 3. Sketch the graph of 𝑓 𝑥 =− 𝑥 3 +2 𝑥 2 +3𝑥
Ex 3. Sketch the graph of 𝑓 𝑥 =− 𝑥 3 +2 𝑥 2 +3𝑥. Be sure to show intercepts, test points (before smallest 𝑥-intercept, between 𝑥-intercepts, and after largest 𝑥-intercept), and end behavior.

25. Ex 3. Sketch the graph of 𝑓 𝑥 =− 𝑥 3 +2 𝑥 2 +3𝑥
Ex 3. Sketch the graph of 𝑓 𝑥 =− 𝑥 3 +2 𝑥 2 +3𝑥. Be sure to show intercepts, test points (before smallest 𝑥-intercept, between 𝑥-intercepts, and after largest 𝑥-intercept), and end behavior.

26. Ex 3. Sketch the graph of 𝑓 𝑥 =− 𝑥 3 +2 𝑥 2 +3𝑥
Ex 3. Sketch the graph of 𝑓 𝑥 =− 𝑥 3 +2 𝑥 2 +3𝑥. Be sure to show intercepts, test points (before smallest 𝑥-intercept, between 𝑥-intercepts, and after largest 𝑥-intercept), and end behavior.

27. Ex 3. Sketch the graph of 𝑓 𝑥 =− 𝑥 3 +2 𝑥 2 +3𝑥
Ex 3. Sketch the graph of 𝑓 𝑥 =− 𝑥 3 +2 𝑥 2 +3𝑥. Be sure to show intercepts, test points (before smallest 𝑥-intercept, between 𝑥-intercepts, and after largest 𝑥-intercept), and end behavior.

28. Ex 3. Sketch the graph of 𝑓 𝑥 =− 𝑥 3 +2 𝑥 2 +3𝑥
Ex 3. Sketch the graph of 𝑓 𝑥 =− 𝑥 3 +2 𝑥 2 +3𝑥. Be sure to show intercepts, test points (before smallest 𝑥-intercept, between 𝑥-intercepts, and after largest 𝑥-intercept), and end behavior.

29. Ex 3. Sketch the graph of 𝑓 𝑥 =− 𝑥 3 +2 𝑥 2 +3𝑥
Ex 3. Sketch the graph of 𝑓 𝑥 =− 𝑥 3 +2 𝑥 2 +3𝑥. Be sure to show intercepts, test points (before smallest 𝑥-intercept, between 𝑥-intercepts, and after largest 𝑥-intercept), and end behavior.

30. Ex 4. Sketch the graph of𝑓 𝑥 =2 𝑥−3 2 𝑥+1 2
Ex 4. Sketch the graph of𝑓 𝑥 =2 𝑥−3 2 𝑥 Be sure to show intercepts, test points (before smallest 𝑥-intercept, between 𝑥-intercepts, and after largest 𝑥-intercept), and end behavior.

31. Ex 4. Sketch the graph of𝑓 𝑥 =2 𝑥−3 2 𝑥+1 2
Ex 4. Sketch the graph of𝑓 𝑥 =2 𝑥−3 2 𝑥 Be sure to show intercepts, test points (before smallest 𝑥-intercept, between 𝑥-intercepts, and after largest 𝑥-intercept), and end behavior.

32. Ex 4. Sketch the graph of𝑓 𝑥 =2 𝑥−3 2 𝑥+1 2
Ex 4. Sketch the graph of𝑓 𝑥 =2 𝑥−3 2 𝑥 Be sure to show intercepts, test points (before smallest 𝑥-intercept, between 𝑥-intercepts, and after largest 𝑥-intercept), and end behavior.

33. Ex 4. Sketch the graph of𝑓 𝑥 =2 𝑥−3 2 𝑥+1 2
Ex 4. Sketch the graph of𝑓 𝑥 =2 𝑥−3 2 𝑥 Be sure to show intercepts, test points (before smallest 𝑥-intercept, between 𝑥-intercepts, and after largest 𝑥-intercept), and end behavior.

34. Ex 4. Sketch the graph of𝑓 𝑥 =2 𝑥−3 2 𝑥+1 2
Ex 4. Sketch the graph of𝑓 𝑥 =2 𝑥−3 2 𝑥 Be sure to show intercepts, test points (before smallest 𝑥-intercept, between 𝑥-intercepts, and after largest 𝑥-intercept), and end behavior.

35. Ex 4. Sketch the graph of𝑓 𝑥 =2 𝑥−3 2 𝑥+1 2
Ex 4. Sketch the graph of𝑓 𝑥 =2 𝑥−3 2 𝑥 Be sure to show intercepts, test points (before smallest 𝑥-intercept, between 𝑥-intercepts, and after largest 𝑥-intercept), and end behavior.

36. Ex 4. Sketch the graph of𝑓 𝑥 =2 𝑥−3 2 𝑥+1 2
Ex 4. Sketch the graph of𝑓 𝑥 =2 𝑥−3 2 𝑥 Be sure to show intercepts, test points (before smallest 𝑥-intercept, between 𝑥-intercepts, and after largest 𝑥-intercept), and end behavior.

37. Multiplicity
For the polynomial 𝑃 𝑥 =2 𝑥− 𝑥−1 4 , the factor 𝑥−5 has multiplicity ___, and the factor 𝑥−1 has multiplicity ___.

38. Multiplicity
For the polynomial 𝑃 𝑥 =2 𝑥− 𝑥−1 4 , the factor 𝑥−5 has multiplicity ___, and the factor 𝑥−1 has multiplicity ___. 𝟑

39. Multiplicity
For the polynomial 𝑃 𝑥 =2 𝑥− 𝑥−1 4 , the factor 𝑥−5 has multiplicity ___, and the factor 𝑥−1 has multiplicity ___. 𝟑 𝟒

40. Multiplicity
If a factor 𝑥−𝑐 has an ______ multiplicity, then the curve will ______________ the 𝑥-axis at 𝑥=𝑐:

41. Multiplicity
If a factor 𝑥−𝑐 has an ______ multiplicity, then the curve will ______________ the 𝑥-axis at 𝑥=𝑐:
odd

42. Multiplicity
If a factor 𝑥−𝑐 has an ______ multiplicity, then the curve will ______________ the 𝑥-axis at 𝑥=𝑐:
odd
pass through

43. Multiplicity
If a factor 𝑥−𝑐 has an ______ multiplicity, then the curve will ______________ the 𝑥-axis at 𝑥=𝑐:

44. Multiplicity
If a factor 𝑥−𝑐 has an ______ multiplicity, then the curve will ______________ the 𝑥-axis at 𝑥=𝑐:
even

45. Multiplicity
If a factor 𝑥−𝑐 has an ______ multiplicity, then the curve will ______________ the 𝑥-axis at 𝑥=𝑐:
even
“bounce” off

46. Ex 5. Based on the graph below, determine if the multiplicities of each zero of 𝑃(𝑥) are even or odd.

47. Ex 5. Based on the graph below, determine if the multiplicities of each zero of 𝑃(𝑥) are even or odd.

48. Ex 5. Based on the graph below, determine if the multiplicities of each zero of 𝑃(𝑥) are even or odd.

49. Ex 5. Based on the graph below, determine if the multiplicities of each zero of 𝑃(𝑥) are even or odd.

50. Ex 5. Based on the graph below, determine if the multiplicities of each zero of 𝑃(𝑥) are even or odd.

51. Ex 5. Based on the graph below, determine if the multiplicities of each zero of 𝑃(𝑥) are even or odd.

52. Note: Every polynomial of degree 𝑛 has at most 𝑛−1 turning points
Note: Every polynomial of degree 𝑛 has at most 𝑛−1 turning points. ex: 𝑃 𝑥 =7 𝑥 4 +3 𝑥 3 −5 𝑥 2 +4𝑥−1 has at most 3 turning points. Also, if a polynomial’s graph has 𝑛 turning points, then the polynomials’ degree is at least 𝑛+1. ex: A polynomial with 5 turning points is at least degree 6.

53. Note: Every polynomial of degree 𝑛 has at most 𝑛−1 turning points
Note: Every polynomial of degree 𝑛 has at most 𝑛−1 turning points. ex: 𝑃 𝑥 =7 𝑥 4 +3 𝑥 3 −5 𝑥 2 +4𝑥−1 has at most 3 turning points. Also, if a polynomial’s graph has 𝑛 turning points, then the polynomials’ degree is at least 𝑛+1. ex: A polynomial with 5 turning points is at least degree 6.

54. Note: Every polynomial of degree 𝑛 has at most 𝑛−1 turning points
Note: Every polynomial of degree 𝑛 has at most 𝑛−1 turning points. ex: 𝑃 𝑥 =7 𝑥 4 +3 𝑥 3 −5 𝑥 2 +4𝑥−1 has at most 3 turning points. Also, if a polynomial’s graph has 𝑛 turning points, then the polynomials’ degree is at least 𝑛+1. ex: A polynomial with 5 turning points is at least degree 6.

55. Note: Every polynomial of degree 𝑛 has at most 𝑛−1 turning points
Note: Every polynomial of degree 𝑛 has at most 𝑛−1 turning points. ex: 𝑃 𝑥 =7 𝑥 4 +3 𝑥 3 −5 𝑥 2 +4𝑥−1 has at most 3 turning points. Also, if a polynomial’s graph has 𝑛 turning points, then the polynomials’ degree is at least 𝑛+1. ex: A polynomial with 5 turning points is at least degree 6.

56. Ex 6. What is the least degree that the polynomial 𝑃 𝑥 shown below could have?

57. Ex 6. What is the least degree that the polynomial 𝑃 𝑥 shown below could have?
3 turning points, so…

58. Ex 6. What is the least degree that the polynomial 𝑃 𝑥 shown below could have?
3 turning points, so…
Least degree: 4