1. Finding Zeros of Polynomials
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2. Divide:
- ( )
- ( )
- ( )

3. Divide:

4. Use Synthetic Division to Divide
x + 2 = 0 1 4 -5 3 -2 -4 18 2 -9 21
x = -2
remainder

5. Use Synthetic Division to Divide
x - 3 = 0 2
-11 3 36 6
-15
-36 -5
-12
x = 3
Factored Form:

6. Factor
x + 1 = 0 1 4
-15
-18 -1 -3 18 3
x = -1

7. Find the zeros of
x = 3
x – 3 = 0 6 -7
-43 30 3 18 33
-30 11
-10

8. A polynomial f(x) has a factor x – k if and only if f(k) = 0.
Factor Theorem
A polynomial f(x) has a factor x – k if 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: p =
factor of constant term a0 q
factor of leading coefficient an

10. List the possible rational zeros15 p =
factor of constant term a0 q
factor of leading coefficient an
1, 3, 5, 15 6
1, 2, 3, 6

11. List the possible rational zeros24 p =
factor of constant term a0 q
factor of leading coefficient an
1, 2, 3, 4, 6, 8, 12, 24 9
1, 3, 9

12. Find all real zeros 3 p = factor of constant term a0 q
factor of leading coefficient an
1, 3 8
1, 2, 4, 8

13. Find all real zeros 8 2 -21 -7 3 1 10 -11 -18 -15 x = 1 Remainder ≠ 0
Therefore, not a factor.

14. Find all real zeros 8 2 -21 -7 3 24 78 171 492 26 57 164 495 x = 3
Remainder ≠ 0
Therefore, not a factor.

15. Find all real zeros 8 2
-21 -7 3
-12 15 9 -3
-10 -6

16. Find all real zeros 1 3 =
1, 3 8
1, 2, 4, 8 4

17. Find all real zeros 4 -5 -3 1 -1 -4

18. Find all real zeros
Rational
Rational
Irrational but Real

19. Look at the graph
≈ 1.618 ≈

20. Look at the graph

21. Find all real zeros 25 p = factor of constant term a0 q
factor of leading coefficient an
1, 5, 25 1 1

22. Find all real zeros 1 -16 -40 -25 -15 -55 -80 5 25 45 9 x = 1 x = 5
-16
-40
-25
-15
-55
-80 5 25 45 9
x = 5
x – 5 = 0

23. Find all real zeros
x = -1 1 5 9 -1 -4 -5 4
x + 1 = 0

24. Find all real zeros
Imaginary -- not Real

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

26. Find all real zeros 12 p = factor of constant term a0 q
factor of leading coefficient an
1, 2, 3, 4, 6, 12 1 1

27. Find all real zeros
x = 1 1 -3 -5 15 4
-12 -2 -7 8 12
x - 1 = 0

28. Find all real zeros
x = 2 1 -2 -7 8 12 2
-14
-12 -6
x - 2 = 0

29. Find all real zeros
x = 3 1 -7 -6 3 9 6 2
x - 3 = 0

30. Find all real zeros

31. Look at the graph
End Behavior?

32. Look at the graph Note: x-min: -5 x-max: 5 x-scale: 1
y-min: -20 y-max: 20 y-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 9 p = factor of constant term a0 q
factor of leading coefficient an
1, 3, 9 1 1

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

37. Find all real zeros 20 p = factor of constant term a0 q
factor of leading coefficient an
1, 2, 4, 5, 10, 20 1 1

38. Find all real zeros 1 -6 7 16 -18 -20 -5 2 18 -8 -2 28 20 -4 -1 14 10-8 -2 28 20 -4 -1 14 10
x = 1
x = 2
x - 2 = 0

39. Find all real zeros 1 -4 -1 14 10 2 -10 8 -2 -5 4 18 5 x = 2 x = -1
x = 2
x = -1
x + 1 = 0

40. Find all real zeros 1 -5 4 10 -1 6
-10 -6
x = -1
x + 1 = 0

41. Find all real zeros

42. End Behavior?

43. Imaginary solutions appear in conjugate pairs.
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. f(x) = (x – 1) (x + 2) (x – 4) f(x) = (x – 1) (x2 – 4x + 2x – 8)
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 !

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

54. Descartes’ Rule of Signs
Let f(x) = anxn + an-1xn a1x + a0 be a polynomial function with real coefficients.
 The number of positive real zeros of f(x) is equal to the number of changes in sign of the coefficients of f(x) or is less than this by an even number.
 The number of negative real zeros of f(x) is equal to the number of changes in sign of the coefficients of f(-x) or is less than this by an even number.

55. Determine the number of positive real zeros.1 2 3
3 sign changes  f(x) has 3 or 1 positive real zero(s).

56. Determine the number of negative real zeros.1 2
2 sign changes  f(x) has 2 or 0 negative real zero(s).

57. Putting it together !  f(x) has 3 or 1 positive real zero(s)
 f(x) has 2 or 0 negative real zero(s)
Positive real zeros
Negative real zeros
Imaginary zeros
Total zeros 3 2 5 1 4

58. Look at the graph Note: x-min: -5 x-max: 5 x-scale: 1
y-min: -30 y-max: 20 y-scale: 5

59. Determine the number of positive real zeros.
0 sign changes  f(x) has 0 positive real zero(s).

60. Determine the number of negative real zeros.1 2 3
3 sign changes  f(x) has 3 or 1 negative real zero(s).

61. Putting it together !  f(x) has 0 positive real zero(s)
 f(x) has 3 or 1 negative real zero(s)
Positive real zeros
Negative real zeros
Imaginary zeros
Total zeros 3 4 7 1 6

62. Look at the graph Note: x-min: -5 x-max: 5 x-scale: 1
y-min: -50 y-max: 50 y-scale: 10

63. Find all real zeros 21 p = factor of constant term a0 q
factor of leading coefficient an
1, 3, 7, 21 18
1, 2, 3, 6, 9, 18
 f(x) has 2 or 0 positive real zero(s)
 f(x) has 1 negative real zero(s)

64. Find all real zeros

65. Find all real zeros upper bound 18 -63 40 21 1 -45 -5 16 2 36 -54 -28
-27
-14 -7 3 54 39 -9 13 60 7
126
441
3367 63
481
3388
upper bound

66. Find all real zeros X X X X X

67. Find all real zeros upper bound 18 -63 40 21 1 -45 -5 16 2 36 -54 -28
Location Principle 18
-63 40 21 1
-45 -5 16 2 36
-54
-28
-27
-14 -7 3 54 39 -9 13 60 7
126
441
3367 63
481
3388
upper bound

68. Find all real zeros X X X X X X X X X X X X

69. Find all real zeros 18
-63 40 21 27
-54
-21
-36
-14 42 14 6