Finding Zeros of Polynomials

1. Finding Zeros of Polynomials Last updated: 12-4-07

2. Divide: - ( ) - ( ) - ( )

3. Divide:

4. Use Synthetic Division to Divide x + 2 = 0 x = -2 remainder

5. Use Synthetic Division to Divide x - 3 = 0 x = 3 Factored Form:

6. Factor x + 1 = 0 x = -1

7. Find the zeros of x = 3 x – 3 = 0

8. Factor Theorem A polynomial f(x) has a factor x – kif and only if f(k) = 0.

9. Rational Zero Theorem If f(x) = anxn + . . . + a1x + a0 has integer coefficients, then every rational zero of f(x) has the following form:

10. List the possible rational zeros 1, 3, 5, 15 15 6 1, 2, 3, 6

11. List the possible rational zeros 1, 2, 3, 4, 6, 8, 12, 24 24 9 1, 3, 9

12. Find all real zeros 1, 3 3 8 1, 2, 4, 8

13. Find all real zeros x = 1 Remainder ≠ 0 Therefore, not a factor.

14. Find all real zeros x = 3 Remainder ≠ 0 Therefore, not a factor.

15. Find all real zeros

16. Find all real zeros 1 4

17. Find all real zeros

18. Find all real zeros Rational Rational Irrational but Real

19. Look at the graph ≈ 1.618 ≈ -0.618

20. Look at the graph

21. Find all real zeros 1, 5, 25 25 1 1

22. Find all real zeros x = 1 x = 5 x – 5 = 0

23. Find all real zeros x = -1 x + 1 = 0

24. Find all real zeros Imaginary -- not Real

25. Look at the graph Note: x-min: -10x-max: 10x-scale: 1 y-min: -250y-max: 100y-scale: 50

26. Find all real zeros 1, 2, 3, 4, 6, 12 12 1 1

27. Find all real zeros x = 1 x - 1 = 0

28. Find all real zeros x = 2 x - 2 = 0

29. Find all real zeros x = 3 x - 3 = 0

30. Find all real zeros

31. Look at the graph End Behavior?

32. Look at the graph Note: x-min: -5x-max: 5x-scale: 1 y-min: -20y-max: 20y-scale: 5

33. Fundamental Theorem of Algebra If f(x) is a polynomial function of degree n where n>0, then the equation f(x) = 0 has at least one solution in the set of complex numbers.

34. Corollary to the Fundamental Theorem of Algebra If f(x) is a polynomial function of degree n where n>0, then the equation f(x) = 0 has exactly n solutions provided each solution repeated twice is counted as 2 solutions, each solution repeated three times is counted as 3 solutions, and so on.

35. Find all real zeros 1, 3, 9 9 1 1

36. Find all real zeros 1 is a multiple root with multiplicity 3 -3 is a multiple root with multiplicity 2

37. Find all real zeros 1, 2, 4, 5, 10, 20 20 1 1

38. Find all real zeros x = 1 x = 2 x - 2 = 0

39. Find all real zeros x = 2 x = -1 x + 1 = 0

40. Find all real zeros x = -1 x + 1 = 0

41. Find all real zeros

42. End Behavior?

43. Key Concepts If f(x) is a polynomial function with real coefficients, and a + bi is an imaginary zero of f(x), then a - bi is also a zero of f(x). Imaginary solutions appear in conjugate pairs.

44. Key Concepts If f(x) is a polynomial function with rational coefficients, and a and b are rational numbers such that ---- is irrational. If --------- is a zero of f(x), then --------- is also a zero of f(x). Irrational solutions containing a square root appear in conjugate pairs.

45. Write a polynomial function f(x) of least degree that has rational coefficients, leading coefficient of 1, and the following zeros  1, -2, 4. x = 1 x = -2 x = 4 x – 1 = 0 x + 2 = 0 x – 4 = 0 f(x) = (x – 1) (x + 2) (x – 4) f(x) = (x – 1) (x2 – 4x + 2x – 8) f(x) = (x – 1) (x2 – 2x – 8) f(x) = x3 – 2x2 – 8x – x2 + 2x + 8 f(x) = x3 – 3x2 – 6x + 8

46. Write a polynomial function f(x) of least degree that has rational coefficients, leading coefficient of 1, and the following zeros  WAIT !

47. Write a polynomial function f(x) of least degree that has rational coefficients, leading coefficient of 1, and the following zeros 

50. Write a polynomial function f(x) of least degree that has rational coefficients, leading coefficient of 1, and the following zeros  WAIT !