Roots and Zeros

1. Section 4-9
Roots and Zeros

2. Essential Questions
• How do you determine the number and type
of roots for a polynomial equation?
• How do you find the zeros of a polynomial
function?

3. Fundamental Theorem of
Algebra

4. Fundamental Theorem of
Algebra
Every polynomial with degree greater than zero
has at least one root in the set of complex
numbers.

5. Example 1
Solve each equation. State the number and type of
roots.
a. x 2
+ 2x − 48 = 0

6. Example 1
Solve each equation. State the number and type of
roots.
a. x 2
+ 2x − 48 = 0
(x + 8)(x − 6) = 0

7. Example 1
Solve each equation. State the number and type of
roots.
a. x 2
+ 2x − 48 = 0
(x + 8)(x − 6) = 0
(x + 8) = 0

8. Example 1
Solve each equation. State the number and type of
roots.
a. x 2
+ 2x − 48 = 0
(x + 8)(x − 6) = 0
(x + 8) = 0 (x − 6) = 0

9. Example 1
Solve each equation. State the number and type of
roots.
a. x 2
+ 2x − 48 = 0
(x + 8)(x − 6) = 0
(x + 8) = 0 (x − 6) = 0
x = −8

10. Example 1
Solve each equation. State the number and type of
roots.
a. x 2
+ 2x − 48 = 0
(x + 8)(x − 6) = 0
(x + 8) = 0 (x − 6) = 0
x = −8 x = 6

11. Example 1
Solve each equation. State the number and type of
roots.
a. x 2
+ 2x − 48 = 0
(x + 8)(x − 6) = 0
(x + 8) = 0 (x − 6) = 0
x = −8 x = 6
There are two real roots.

12. Example 1
Solve each equation. State the number and type of
roots.
b. y 4
− 256 = 0

13. Example 1
Solve each equation. State the number and type of
roots.
b. y 4
− 256 = 0
(y 2
+16)(y 2
−16) = 0

14. Example 1
Solve each equation. State the number and type of
roots.
b. y 4
− 256 = 0
(y 2
+16)(y 2
−16) = 0
(y + 4i )(y − 4i )(y + 4)(y − 4) = 0

15. Example 1
Solve each equation. State the number and type of
roots.
b. y 4
− 256 = 0
(y 2
+16)(y 2
−16) = 0
(y + 4i ) = 0
(y + 4i )(y − 4i )(y + 4)(y − 4) = 0

16. Example 1
Solve each equation. State the number and type of
roots.
b. y 4
− 256 = 0
(y 2
+16)(y 2
−16) = 0
(y + 4i ) = 0
(y + 4i )(y − 4i )(y + 4)(y − 4) = 0
(y − 4i ) = 0

17. Example 1
Solve each equation. State the number and type of
roots.
b. y 4
− 256 = 0
(y 2
+16)(y 2
−16) = 0
(y + 4i ) = 0
(y + 4i )(y − 4i )(y + 4)(y − 4) = 0
(y − 4i ) = 0 (y + 4) = 0

18. Example 1
Solve each equation. State the number and type of
roots.
b. y 4
− 256 = 0
(y 2
+16)(y 2
−16) = 0
(y + 4i ) = 0
(y + 4i )(y − 4i )(y + 4)(y − 4) = 0
(y − 4i ) = 0 (y + 4) = 0 (y − 4) = 0

19. Example 1
Solve each equation. State the number and type of
roots.
b. y 4
− 256 = 0
(y 2
+16)(y 2
−16) = 0
(y + 4i ) = 0
(y + 4i )(y − 4i )(y + 4)(y − 4) = 0
(y − 4i ) = 0 (y + 4) = 0 (y − 4) = 0
y = −4i

20. Example 1
Solve each equation. State the number and type of
roots.
b. y 4
− 256 = 0
(y 2
+16)(y 2
−16) = 0
(y + 4i ) = 0
(y + 4i )(y − 4i )(y + 4)(y − 4) = 0
(y − 4i ) = 0 (y + 4) = 0 (y − 4) = 0
y = −4i y = 4i

21. Example 1
Solve each equation. State the number and type of
roots.
b. y 4
− 256 = 0
(y 2
+16)(y 2
−16) = 0
(y + 4i ) = 0
(y + 4i )(y − 4i )(y + 4)(y − 4) = 0
(y − 4i ) = 0 (y + 4) = 0 (y − 4) = 0
y = −4i y = 4i y = −4

22. Example 1
Solve each equation. State the number and type of
roots.
b. y 4
− 256 = 0
(y 2
+16)(y 2
−16) = 0
(y + 4i ) = 0
(y + 4i )(y − 4i )(y + 4)(y − 4) = 0
(y − 4i ) = 0 (y + 4) = 0 (y − 4) = 0
y = −4i y = 4i y = −4 y = 4

23. Example 1
Solve each equation. State the number and type of
roots.
b. y 4
− 256 = 0
(y 2
+16)(y 2
−16) = 0
(y + 4i ) = 0
There are two imaginary root and two real roots.
(y + 4i )(y − 4i )(y + 4)(y − 4) = 0
(y − 4i ) = 0 (y + 4) = 0 (y − 4) = 0
y = −4i y = 4i y = −4 y = 4

24. Corollary to the Fundamental
Theorem of Algebra

25. Corollary to the Fundamental
Theorem of Algebra
A polynomial equation of degree n has exactly n
roots in the set of complex numbers, including
double roots.

26. Descartes’ Rule of Signs
Let be a polynomial
function with real coefficients. Then:
P(x ) = anxn
+...+ a1x + a0

27. Descartes’ Rule of Signs
Let be a polynomial
function with real coefficients. Then:
P(x ) = anxn
+...+ a1x + a0
• The number of positive real zeros of P(x) is
the same as the number of changes in sign of
the coefficients of the terms, or is less than
this by an even number

28. Descartes’ Rule of Signs
Let be a polynomial
function with real coefficients. Then:
P(x ) = anxn
+...+ a1x + a0
• The number of positive real zeros of P(x) is
the same as the number of changes in sign of
the coefficients of the terms, or is less than
this by an even number
• The number of negative real zeros of P(x) is
the same as the number of changes in sign of
the coefficients of the terms of P(−x), or is
less than this by an even number

29. Example 2
p(x ) = −x 6
+ 4x 3
− 2x 2
− x −1
State the possible number of positive real zeros,
negative real zeros, and imaginary zeros of

30. Example 2
p(x ) = −x 6
+ 4x 3
− 2x 2
− x −1
State the possible number of positive real zeros,
negative real zeros, and imaginary zeros of

31. Example 2
p(x ) = −x 6
+ 4x 3
− 2x 2
− x −1
State the possible number of positive real zeros,
negative real zeros, and imaginary zeros of
Yes

32. Example 2
p(x ) = −x 6
+ 4x 3
− 2x 2
− x −1
State the possible number of positive real zeros,
negative real zeros, and imaginary zeros of
Yes

33. Example 2
p(x ) = −x 6
+ 4x 3
− 2x 2
− x −1
State the possible number of positive real zeros,
negative real zeros, and imaginary zeros of
Yes Yes

34. Example 2
p(x ) = −x 6
+ 4x 3
− 2x 2
− x −1
State the possible number of positive real zeros,
negative real zeros, and imaginary zeros of
Yes Yes

35. Example 2
p(x ) = −x 6
+ 4x 3
− 2x 2
− x −1
State the possible number of positive real zeros,
negative real zeros, and imaginary zeros of
Yes Yes No

36. Example 2
p(x ) = −x 6
+ 4x 3
− 2x 2
− x −1
State the possible number of positive real zeros,
negative real zeros, and imaginary zeros of
Yes Yes No

37. Example 2
p(x ) = −x 6
+ 4x 3
− 2x 2
− x −1
State the possible number of positive real zeros,
negative real zeros, and imaginary zeros of
Yes Yes No No

38. Example 2
p(x ) = −x 6
+ 4x 3
− 2x 2
− x −1
State the possible number of positive real zeros,
negative real zeros, and imaginary zeros of
Yes Yes No No
There are two sign changes, so there are either 2 or 0
positive zeros

39. Example 2
State the possible number of positive real zeros,
negative real zeros, and imaginary zeros of
p(−x ) = −(−x )6
+ 4(−x )3
− 2(−x )2
− (−x )−1

40. Example 2
State the possible number of positive real zeros,
negative real zeros, and imaginary zeros of
p(−x ) = −(−x )6
+ 4(−x )3
− 2(−x )2
− (−x )−1
p(−x ) = −x 6
− 4x 3
− 2x 2
+ x −1

41. Example 2
State the possible number of positive real zeros,
negative real zeros, and imaginary zeros of
p(−x ) = −(−x )6
+ 4(−x )3
− 2(−x )2
− (−x )−1
p(−x ) = −x 6
− 4x 3
− 2x 2
+ x −1

42. Example 2
State the possible number of positive real zeros,
negative real zeros, and imaginary zeros of
p(−x ) = −(−x )6
+ 4(−x )3
− 2(−x )2
− (−x )−1
No
p(−x ) = −x 6
− 4x 3
− 2x 2
+ x −1

43. Example 2
State the possible number of positive real zeros,
negative real zeros, and imaginary zeros of
p(−x ) = −(−x )6
+ 4(−x )3
− 2(−x )2
− (−x )−1
No
p(−x ) = −x 6
− 4x 3
− 2x 2
+ x −1

44. Example 2
State the possible number of positive real zeros,
negative real zeros, and imaginary zeros of
p(−x ) = −(−x )6
+ 4(−x )3
− 2(−x )2
− (−x )−1
No No
p(−x ) = −x 6
− 4x 3
− 2x 2
+ x −1

45. Example 2
State the possible number of positive real zeros,
negative real zeros, and imaginary zeros of
p(−x ) = −(−x )6
+ 4(−x )3
− 2(−x )2
− (−x )−1
No No
p(−x ) = −x 6
− 4x 3
− 2x 2
+ x −1

46. Example 2
State the possible number of positive real zeros,
negative real zeros, and imaginary zeros of
p(−x ) = −(−x )6
+ 4(−x )3
− 2(−x )2
− (−x )−1
No No Yes
p(−x ) = −x 6
− 4x 3
− 2x 2
+ x −1

47. Example 2
State the possible number of positive real zeros,
negative real zeros, and imaginary zeros of
p(−x ) = −(−x )6
+ 4(−x )3
− 2(−x )2
− (−x )−1
No No Yes
p(−x ) = −x 6
− 4x 3
− 2x 2
+ x −1

48. Example 2
State the possible number of positive real zeros,
negative real zeros, and imaginary zeros of
p(−x ) = −(−x )6
+ 4(−x )3
− 2(−x )2
− (−x )−1
No No Yes Yes
p(−x ) = −x 6
− 4x 3
− 2x 2
+ x −1

49. Example 2
State the possible number of positive real zeros,
negative real zeros, and imaginary zeros of
There are two sign changes, so there are either 2 or 0
negative zeros
p(−x ) = −(−x )6
+ 4(−x )3
− 2(−x )2
− (−x )−1
No No Yes Yes
p(−x ) = −x 6
− 4x 3
− 2x 2
+ x −1

50. Example 2
p(x ) = −x 6
+ 4x 3
− 2x 2
− x −1
State the possible number of positive real zeros,
negative real zeros, and imaginary zeros of

51. Example 2
p(x ) = −x 6
+ 4x 3
− 2x 2
− x −1
State the possible number of positive real zeros,
negative real zeros, and imaginary zeros of
+ zeros − zeros
imaginary
zeros
total
zeros

52. Example 2
p(x ) = −x 6
+ 4x 3
− 2x 2
− x −1
State the possible number of positive real zeros,
negative real zeros, and imaginary zeros of
+ zeros − zeros
imaginary
zeros
total
zeros
2

53. Example 2
p(x ) = −x 6
+ 4x 3
− 2x 2
− x −1
State the possible number of positive real zeros,
negative real zeros, and imaginary zeros of
+ zeros − zeros
imaginary
zeros
total
zeros
2 2

54. Example 2
p(x ) = −x 6
+ 4x 3
− 2x 2
− x −1
State the possible number of positive real zeros,
negative real zeros, and imaginary zeros of
+ zeros − zeros
imaginary
zeros
total
zeros
2 2 2

55. Example 2
p(x ) = −x 6
+ 4x 3
− 2x 2
− x −1
State the possible number of positive real zeros,
negative real zeros, and imaginary zeros of
+ zeros − zeros
imaginary
zeros
total
zeros
2 2 2 6

56. Example 2
p(x ) = −x 6
+ 4x 3
− 2x 2
− x −1
State the possible number of positive real zeros,
negative real zeros, and imaginary zeros of
+ zeros − zeros
imaginary
zeros
total
zeros
2 2 2 6
2

57. Example 2
p(x ) = −x 6
+ 4x 3
− 2x 2
− x −1
State the possible number of positive real zeros,
negative real zeros, and imaginary zeros of
+ zeros − zeros
imaginary
zeros
total
zeros
2 2 2 6
2 0

58. Example 2
p(x ) = −x 6
+ 4x 3
− 2x 2
− x −1
State the possible number of positive real zeros,
negative real zeros, and imaginary zeros of
+ zeros − zeros
imaginary
zeros
total
zeros
2 2 2 6
2 0 4

59. Example 2
p(x ) = −x 6
+ 4x 3
− 2x 2
− x −1
State the possible number of positive real zeros,
negative real zeros, and imaginary zeros of
+ zeros − zeros
imaginary
zeros
total
zeros
2 2 2 6
2 0 4 6

60. Example 2
p(x ) = −x 6
+ 4x 3
− 2x 2
− x −1
State the possible number of positive real zeros,
negative real zeros, and imaginary zeros of
+ zeros − zeros
imaginary
zeros
total
zeros
2 2 2 6
2 0 4 6
0

61. Example 2
p(x ) = −x 6
+ 4x 3
− 2x 2
− x −1
State the possible number of positive real zeros,
negative real zeros, and imaginary zeros of
+ zeros − zeros
imaginary
zeros
total
zeros
2 2 2 6
2 0 4 6
0 2

62. Example 2
p(x ) = −x 6
+ 4x 3
− 2x 2
− x −1
State the possible number of positive real zeros,
negative real zeros, and imaginary zeros of
+ zeros − zeros
imaginary
zeros
total
zeros
2 2 2 6
2 0 4 6
0 2 4

63. Example 2
p(x ) = −x 6
+ 4x 3
− 2x 2
− x −1
State the possible number of positive real zeros,
negative real zeros, and imaginary zeros of
+ zeros − zeros
imaginary
zeros
total
zeros
2 2 2 6
2 0 4 6
0 2 4 6

64. Example 2
p(x ) = −x 6
+ 4x 3
− 2x 2
− x −1
State the possible number of positive real zeros,
negative real zeros, and imaginary zeros of
+ zeros − zeros
imaginary
zeros
total
zeros
2 2 2 6
2 0 4 6
0 2 4 6
0

65. Example 2
p(x ) = −x 6
+ 4x 3
− 2x 2
− x −1
State the possible number of positive real zeros,
negative real zeros, and imaginary zeros of
+ zeros − zeros
imaginary
zeros
total
zeros
2 2 2 6
2 0 4 6
0 2 4 6
0 0

66. Example 2
p(x ) = −x 6
+ 4x 3
− 2x 2
− x −1
State the possible number of positive real zeros,
negative real zeros, and imaginary zeros of
+ zeros − zeros
imaginary
zeros
total
zeros
2 2 2 6
2 0 4 6
0 2 4 6
0 0 6

67. Example 2
p(x ) = −x 6
+ 4x 3
− 2x 2
− x −1
State the possible number of positive real zeros,
negative real zeros, and imaginary zeros of
+ zeros − zeros
imaginary
zeros
total
zeros
2 2 2 6
2 0 4 6
0 2 4 6
0 0 6 6

68. Example 2
p(x ) = −x 6
+ 4x 3
− 2x 2
− x −1
State the possible number of positive real zeros,
negative real zeros, and imaginary zeros of

69. Example 2
p(x ) = −x 6
+ 4x 3
− 2x 2
− x −1
State the possible number of positive real zeros,
negative real zeros, and imaginary zeros of
There are either 2 or 0 real positive roots.

70. Example 2
p(x ) = −x 6
+ 4x 3
− 2x 2
− x −1
State the possible number of positive real zeros,
negative real zeros, and imaginary zeros of
There are either 2 or 0 real positive roots.
There are either 2 or 0 real negative roots.

71. Example 2
p(x ) = −x 6
+ 4x 3
− 2x 2
− x −1
State the possible number of positive real zeros,
negative real zeros, and imaginary zeros of
There are either 2 or 0 real positive roots.
There are either 2 or 0 real negative roots.
There are either 6, 4, or 2 real negative roots.

72. Example 3
Find all of the zeros of
f (x ) = x 3
− x 2
+ 2x + 4

73. Example 3
Find all of the zeros of
f (x ) = x 3
− x 2
+ 2x + 4

74. Example 3
Find all of the zeros of
f (x ) = x 3
− x 2
+ 2x + 4
Yes

75. Example 3
Find all of the zeros of
f (x ) = x 3
− x 2
+ 2x + 4
Yes

76. Example 3
Find all of the zeros of
f (x ) = x 3
− x 2
+ 2x + 4
Yes Yes

77. Example 3
Find all of the zeros of
f (x ) = x 3
− x 2
+ 2x + 4
Yes Yes

78. Example 3
Find all of the zeros of
f (x ) = x 3
− x 2
+ 2x + 4
Yes Yes No

79. Example 3
Find all of the zeros of
f (x ) = x 3
− x 2
+ 2x + 4
Yes Yes No
There are either 2 or 0 real positive zeros

80. Example 3
Find all of the zeros of
f (x ) = x 3
− x 2
+ 2x + 4
Yes Yes No
There are either 2 or 0 real positive zeros
f (−x ) = (−x )3
− (−x )2
+ 2(−x )+ 4

81. Example 3
Find all of the zeros of
f (x ) = x 3
− x 2
+ 2x + 4
Yes Yes No
There are either 2 or 0 real positive zeros
f (−x ) = (−x )3
− (−x )2
+ 2(−x )+ 4
f (−x ) = −x 3
− x 2
− 2x + 4

82. Example 3
Find all of the zeros of
f (x ) = x 3
− x 2
+ 2x + 4
Yes Yes No
There are either 2 or 0 real positive zeros
f (−x ) = (−x )3
− (−x )2
+ 2(−x )+ 4
f (−x ) = −x 3
− x 2
− 2x + 4

83. Example 3
Find all of the zeros of
f (x ) = x 3
− x 2
+ 2x + 4
Yes Yes No
There are either 2 or 0 real positive zeros
f (−x ) = (−x )3
− (−x )2
+ 2(−x )+ 4
No
f (−x ) = −x 3
− x 2
− 2x + 4

84. Example 3
Find all of the zeros of
f (x ) = x 3
− x 2
+ 2x + 4
Yes Yes No
There are either 2 or 0 real positive zeros
f (−x ) = (−x )3
− (−x )2
+ 2(−x )+ 4
No
f (−x ) = −x 3
− x 2
− 2x + 4

85. Example 3
Find all of the zeros of
f (x ) = x 3
− x 2
+ 2x + 4
Yes Yes No
There are either 2 or 0 real positive zeros
f (−x ) = (−x )3
− (−x )2
+ 2(−x )+ 4
No No
f (−x ) = −x 3
− x 2
− 2x + 4

86. Example 3
Find all of the zeros of
f (x ) = x 3
− x 2
+ 2x + 4
Yes Yes No
There are either 2 or 0 real positive zeros
f (−x ) = (−x )3
− (−x )2
+ 2(−x )+ 4
No No
f (−x ) = −x 3
− x 2
− 2x + 4

87. Example 3
Find all of the zeros of
f (x ) = x 3
− x 2
+ 2x + 4
Yes Yes No
There are either 2 or 0 real positive zeros
f (−x ) = (−x )3
− (−x )2
+ 2(−x )+ 4
No No Yes
f (−x ) = −x 3
− x 2
− 2x + 4

88. Example 3
Find all of the zeros of
f (x ) = x 3
− x 2
+ 2x + 4
Yes Yes No
There are either 2 or 0 real positive zeros
f (−x ) = (−x )3
− (−x )2
+ 2(−x )+ 4
No No Yes
There is 1 real negative zero
f (−x ) = −x 3
− x 2
− 2x + 4

89. Example 3
f (x ) = x 3
− x 2
+ 2x + 4

90. Example 3
+ zeros − zeros
imaginary
zeros
total
zeros
f (x ) = x 3
− x 2
+ 2x + 4

91. Example 3
+ zeros − zeros
imaginary
zeros
total
zeros
2
f (x ) = x 3
− x 2
+ 2x + 4

92. Example 3
+ zeros − zeros
imaginary
zeros
total
zeros
2 1
f (x ) = x 3
− x 2
+ 2x + 4

93. Example 3
+ zeros − zeros
imaginary
zeros
total
zeros
2 1 0
f (x ) = x 3
− x 2
+ 2x + 4

94. Example 3
+ zeros − zeros
imaginary
zeros
total
zeros
2 1 0 3
f (x ) = x 3
− x 2
+ 2x + 4

95. Example 3
+ zeros − zeros
imaginary
zeros
total
zeros
2 1 0 3
0
f (x ) = x 3
− x 2
+ 2x + 4

96. Example 3
+ zeros − zeros
imaginary
zeros
total
zeros
2 1 0 3
0 1
f (x ) = x 3
− x 2
+ 2x + 4

97. Example 3
+ zeros − zeros
imaginary
zeros
total
zeros
2 1 0 3
0 1 2
f (x ) = x 3
− x 2
+ 2x + 4

98. Example 3
+ zeros − zeros
imaginary
zeros
total
zeros
2 1 0 3
0 1 2 3
f (x ) = x 3
− x 2
+ 2x + 4

99. Example 3
Find all of the zeros of
f (x ) = x 3
− x 2
+ 2x + 4

100. Example 3
Find all of the zeros of
f (x ) = x 3
− x 2
+ 2x + 4
Not factorable, so find a root with the calculator.

101. Example 3
Find all of the zeros of
f (x ) = x 3
− x 2
+ 2x + 4
Not factorable, so find a root with the calculator.

102. Example 3
Find all of the zeros of
f (x ) = x 3
− x 2
+ 2x + 4
Not factorable, so find a root with the calculator.
First zero: x = −1

103. Example 3
Find all of the zeros of
f (x ) = x 3
− x 2
+ 2x + 4
Not factorable, so find a root with the calculator.
First zero: x = −1
1 −1 2 4−1

104. Example 3
Find all of the zeros of
f (x ) = x 3
− x 2
+ 2x + 4
Not factorable, so find a root with the calculator.
First zero: x = −1
1 −1 2 4−1
1

105. Example 3
Find all of the zeros of
f (x ) = x 3
− x 2
+ 2x + 4
Not factorable, so find a root with the calculator.
First zero: x = −1
1 −1 2 4−1
1
−1

106. Example 3
Find all of the zeros of
f (x ) = x 3
− x 2
+ 2x + 4
Not factorable, so find a root with the calculator.
First zero: x = −1
1 −1 2 4−1
1
−1
−2

107. Example 3
Find all of the zeros of
f (x ) = x 3
− x 2
+ 2x + 4
Not factorable, so find a root with the calculator.
First zero: x = −1
1 −1 2 4−1
1
−1
−2
2

108. Example 3
Find all of the zeros of
f (x ) = x 3
− x 2
+ 2x + 4
Not factorable, so find a root with the calculator.
First zero: x = −1
1 −1 2 4−1
1
−1
−2
2
4

109. Example 3
Find all of the zeros of
f (x ) = x 3
− x 2
+ 2x + 4
Not factorable, so find a root with the calculator.
First zero: x = −1
1 −1 2 4−1
1
−1
−2
2
4
−4

110. Example 3
Find all of the zeros of
f (x ) = x 3
− x 2
+ 2x + 4
Not factorable, so find a root with the calculator.
First zero: x = −1
1 −1 2 4−1
1
−1
−2
2
4
−4
0

111. Example 3
Find all of the zeros of
f (x ) = x 3
− x 2
+ 2x + 4
Not factorable, so find a root with the calculator.
First zero: x = −1
1 −1 2 4−1
1
−1
−2
2
4
−4
0

112. Example 3
Find all of the zeros of
f (x ) = x 3
− x 2
+ 2x + 4
Not factorable, so find a root with the calculator.
First zero: x = −1
1 −1 2 4−1
1
−1
−2
2
4
−4
0
(x +1)(x 2
− 2x + 4)

113. Example 3
Find all of the zeros of
f (x ) = x 3
− x 2
+ 2x + 4
Since we have one negative root, the other roots are
imaginary.
(x +1)(x 2
− 2x + 4)

114. Example 3
Find all of the zeros of
f (x ) = x 3
− x 2
+ 2x + 4
Since we have one negative root, the other roots are
imaginary.
(x +1)(x 2
− 2x + 4)
x =
−b ± b2
− 4ac
2a

115. Example 3
Find all of the zeros of
f (x ) = x 3
− x 2
+ 2x + 4
Since we have one negative root, the other roots are
imaginary.
(x +1)(x 2
− 2x + 4)
x =
−b ± b2
− 4ac
2a
x =
2 ± (−2)2
− 4(1)(4)
2(1)

116. Example 3
Find all of the zeros of
f (x ) = x 3
− x 2
+ 2x + 4
Since we have one negative root, the other roots are
imaginary.
(x +1)(x 2
− 2x + 4)
x =
−b ± b2
− 4ac
2a
x =
2 ± (−2)2
− 4(1)(4)
2(1)
x =
2 ± 4 −16
2

117. Example 3
Find all of the zeros of
f (x ) = x 3
− x 2
+ 2x + 4
Since we have one negative root, the other roots are
imaginary.
(x +1)(x 2
− 2x + 4)
x =
−b ± b2
− 4ac
2a
x =
2 ± (−2)2
− 4(1)(4)
2(1)
x =
2 ± 4 −16
2
x =
2 ± −12
2

118. Example 3
Find all of the zeros of
f (x ) = x 3
− x 2
+ 2x + 4
Since we have one negative root, the other roots are
imaginary.
(x +1)(x 2
− 2x + 4)
x =
−b ± b2
− 4ac
2a
x =
2 ± (−2)2
− 4(1)(4)
2(1)
x =
2 ± 4 −16
2
x =
2 ± −12
2
x =
2 ± 2i 3
2

119. Example 3
Find all of the zeros of
f (x ) = x 3
− x 2
+ 2x + 4
Since we have one negative root, the other roots are
imaginary.
(x +1)(x 2
− 2x + 4)
x =
−b ± b2
− 4ac
2a
x =
2 ± (−2)2
− 4(1)(4)
2(1)
x =
2 ± 4 −16
2
x =
2 ± −12
2
x =
2 ± 2i 3
2
x = 1± i 3

120. Complex Conjugates
Theorem

121. Complex Conjugates
Theorem
Let a and b be real numbers and b ≠ 0. If a + bi
is a zero of a polynomial with real coefficients,
then a − bi is also a zero of the function.

122. Example 4
4 − i
Write a polynomial function of least degree with integer
coefficients, the zeros of which include 4 and .

123. Example 4
4 − i
Write a polynomial function of least degree with integer
coefficients, the zeros of which include 4 and .
x = 4

124. Example 4
4 − i
Write a polynomial function of least degree with integer
coefficients, the zeros of which include 4 and .
x = 4 x = 4 − i

125. Example 4
4 − i
Write a polynomial function of least degree with integer
coefficients, the zeros of which include 4 and .
x = 4 x = 4 + ix = 4 − i

126. Example 4
4 − i
Write a polynomial function of least degree with integer
coefficients, the zeros of which include 4 and .
x = 4 x = 4 + ix = 4 − i
(x − 4) = 0

127. Example 4
4 − i
Write a polynomial function of least degree with integer
coefficients, the zeros of which include 4 and .
x = 4 x = 4 + ix = 4 − i
(x − 4) = 0 [x − (4 − i )] = 0

128. Example 4
4 − i
Write a polynomial function of least degree with integer
coefficients, the zeros of which include 4 and .
x = 4 x = 4 + ix = 4 − i
(x − 4) = 0 [x − (4 − i )] = 0 [x − (4 + i )] = 0

129. Example 4
4 − i
Write a polynomial function of least degree with integer
coefficients, the zeros of which include 4 and .
x = 4 x = 4 + ix = 4 − i
(x − 4) = 0 [x − (4 − i )] = 0 [x − (4 + i )] = 0
(x − 4)[x − (4 − i )][x − (4 + i )] = 0

130. Example 4
4 − i
Write a polynomial function of least degree with integer
coefficients, the zeros of which include 4 and .
x = 4 x = 4 + ix = 4 − i
(x − 4) = 0 [x − (4 − i )] = 0 [x − (4 + i )] = 0
(x − 4)[x − (4 − i )][x − (4 + i )] = 0
(x − 4)[(x − 4)+ i )][(x − 4)− i )] = 0

131. Example 4
4 − i
Write a polynomial function of least degree with integer
coefficients, the zeros of which include 4 and .
x = 4 x = 4 + ix = 4 − i
(x − 4) = 0 [x − (4 − i )] = 0 [x − (4 + i )] = 0
(x − 4)[x − (4 − i )][x − (4 + i )] = 0
(x − 4)[(x − 4)+ i )][(x − 4)− i )] = 0
(x − 4)[(x − 4)2
− i 2
)] = 0

132. Example 4
4 − i
Write a polynomial function of least degree with integer
coefficients, the zeros of which include 4 and .
x = 4 x = 4 + ix = 4 − i
(x − 4) = 0 [x − (4 − i )] = 0 [x − (4 + i )] = 0
(x − 4)[x − (4 − i )][x − (4 + i )] = 0
(x − 4)[(x − 4)+ i )][(x − 4)− i )] = 0
(x − 4)[(x − 4)2
− i 2
)] = 0
(x − 4)[x 2
− 8x +16 − (−1)] = 0

133. Example 4
4 − i
Write a polynomial function of least degree with integer
coefficients, the zeros of which include 4 and .
(x − 4)[x 2
− 8x +16 − (−1)] = 0

134. Example 4
4 − i
Write a polynomial function of least degree with integer
coefficients, the zeros of which include 4 and .
(x − 4)[x 2
− 8x +16 − (−1)] = 0
(x − 4)(x 2
− 8x +17) = 0

135. Example 4
4 − i
Write a polynomial function of least degree with integer
coefficients, the zeros of which include 4 and .
(x − 4)[x 2
− 8x +16 − (−1)] = 0
(x − 4)(x 2
− 8x +17) = 0
x 3
− 8x 2
+17x − 4x 2
+ 32x − 68 = 0

136. Example 4
4 − i
Write a polynomial function of least degree with integer
coefficients, the zeros of which include 4 and .
(x − 4)[x 2
− 8x +16 − (−1)] = 0
(x − 4)(x 2
− 8x +17) = 0
x 3
− 8x 2
+17x − 4x 2
+ 32x − 68 = 0
x 3
−12x 2
+ 49x − 68 = 0