1. Remainder and Factor Theorems Unit 11

2. Definitions The real number, r, is a zero of f(x) iff:  r is a solution, or root, of f(x)=0  x-r is a factor of the expression that defines f (f(r)=0)  when the expression is divided by x- r, the remainder is 0  r is an x-intercept of the graph of f.

3. Remainder Theorem If the polynomial expression that defines the function of P is divided by x-r, then the remainder is the number P(r).

4. Factor Theorem x-r is a factor of the polynomial expression that defines the function P iff r is a solution of P(x)=0. That is, if P(r)=0.

5. Integer Roots Unit 11

6. Page 464, #42-44 42) 43) 44)

7. Warm Up Find the polynomial P(x) in the standard form that has roots of x = {-3, -1, 1} and P(0) = 9.

8. Quiz Factor & Remainder Theorem.

9. Example As the first step in creating a graph of the polynomial, find all x-intercepts of the polynomial :.

10. Rational Root Theorem Let P be a polynomial function with integer coefficients in standard form. If is a root of P(x) = 0, then  p is a factor of the constant term of P and  q is a factor of the leading coefficient of P.

11. Determine the number of roots. List all factors of the constant term. List all factors of the leading coefficient. List all the possible roots. Test each possible root to find the zeros of each polynomial.

12. Examples 1.) 2.)

13. Examples 3.) 4.)

14. Assignment Worksheet #1, 1-5

15. Show What You Know

16. Rational Roots Unit 11

17. Warm Up List all possible roots and use them to find the zeros of the polynomial:

18. Worksheet #1, 1-5 1) -6, -1, 1 2) -3, -2, 2 3) -4, 2 (multiplicity 2) 4) -1 (multiplicity 2), 2 5) -3, -1, 2, 3

19. Rational Root Theorem Let P be a polynomial function with integer coefficients in standard form. If is a root of P(x) = 0, then  p is a factor of the constant term of P and  q is a factor of the leading coefficient of P.

20. Determine the number of roots. List all factors of the constant term. List all factors of the leading coefficient. List all the possible roots. Test each possible root (using substitution or synthetic division) to find the zeros of each polynomial.

21. Examples 1.) 2.)

22. Assignment Worksheet #2, 1-16

23. Show What You Know

24. Rational Roots Unit 11

25. Warm Up List all possible roots and use them to find the zeros of the polynomial:

26. Worksheet #2, 1-16 1) 1, 1/3 2) 1, 2, 4, 8, 16, 32, 64 3) 1, 2, 5, 10 4) 1, 2, 4, 8, 1/5, 2/5, 4/5, 8/5 5) 1, 5, 25, ½, 5/2, 25/2, ¼, 5/4, 25/4 6) 1, 3, 7, 21, 1/5, 3/5, 7/5, 21/5 7) 1, 3, 9, 27 8) 1, 7, ½, 7/2

27. Worksheet #2, 1-16 9) x={1 (multiplicity 2), -3} 10) x={1 (multiplicity 2), 11} 11) x={-1 (multiplicity 2), -2} 12) x={-1, 1/5, -5} 13) x={1 (multiplicity 2), ¼} 14) x={-1, 1/3, -3} 15) x={1 (multiplicity 2), 1/5, 7} 16) x={-1 (multiplicity 2), 1/3, 5}

28. Quiz Integer and Rational Roots

29. Graphing Polynomials Unit 11

30. Warm Up Determine the number of roots. Then find the roots of the polynomial.

31. Critical Thinking In the process of solving you test 1, 2, 5, and 10 as possible zeros and determine that none of them are actual zeros. You then discover that -5/2 is a zero. You calculate the depressed polynomial to be Do you need to test 1, 2, 5, and 10 again? Why or why not?

32. End Behavior What happens to a polynomial function as its x- values get very small and very large is called the end behavior of the function.

33. End Behavior f(x)=ax n +… a > 0a < 0 leftrightleftright n is even n is odd

34. Leading Coefficient > 0

35. Leading Coefficient < 0

36. End Behavior f(x)=ax n +… a > 0a < 0 leftrightleftright n is even increasedecrease n is odd increasedecrease

37. Highest Exponent is Odd

38. Highest Exponent is Even

39. End Behavior f(x)=ax n +… a > 0a < 0 leftrightleftright n is even increase decrease n is odd decreaseincrease decrease

40. Examples Sketch the graph of each polynomial. 1.) 2.) 3.)

41. Assignment Worksheet 3, #1-8

42. Exit Survey Which of the following is the graph of ? A. B. C. D. B.

43. Polynomial Review Unit 11

44. Warm Up  Sketch a graph of the polynomial:

45. A=True B=False a) If f(-5)=0, then (x-5) is a factor of f(x). b) If x=9 is a root of f(x), then (x-9) is a factor of f(x). c) If the polynomial f(x) is synthetically divided by (x-4) and the remainder is 0, then f(4)=0. Example #1

46. Determine if (x+1) is a factor of the polynomial: A=Yes B=No Example #2

47. Example #3 Find the polynomial, in factored form, with the roots x={-2,2,4} and f(1)=18. A B C D

48. Example #4 How many roots will the function have? List all the possible rational roots. Perform the synthetic division. Write the polynomial in its factored form with each factor having only integer coefficients. Write the roots of the polynomial. Sketch the graph.

49. Assignment Review Sheet