3.6 Fundamental Theorem of Algebra
Objectives:
N.CN.9: Know the Fundamental Theorem of Algebra; show that it is true for quadratic
polynomials.
N.CN.8: Extend polynomial identities to the complex numbers. For example, rewrite x 2 + 4
as (x + 2i)(x – 2i).
N.CN.7: Solve quadratic equations with real coefficients that have complex solutions.
A.APR.2: Know and apply the Remainder Theorem …
For the board: You will be able to use the Fundamental Theorem of Algebra and its corollary to write a
polynomial equation of least degree with given roots.
You will be able to identify all of the roots of a polynomial equation.
Bell Work 3.6:
Solve each of the following polynomials using factoring.
1. 4x5 – 8x4 – 32x3 = 0
2. x4 + 16 = 17x2
Anticipatory Set:
The following statements are equivalent:
 A real number r is a root of the polynomial equation P(x) = 0.
3 is a root of the polynomial equation x2 – 2x – 3 = 0
 P(r) = 0
P(3) = O for P(x) = x2 – 2x – 3.
 r is an x-intercept of the graph of P(x).
(3, 0) is an x-intercept of y = x2 – 2x – 3.
 x – r is a factor of P(x).
x – 3 is a factor of P(x) = x2 – 2x – 3.
 When you divide the rule for P(x) by x – r, the remainder is 0.
(x2 – 2x – 3)  (x – 3) has a remainder of 0.
 r is a zero of P(x).
(3, 0) is a zero of P(x) = x2 – 2x – 3.
Open the book to page 189 and read example 1.
Example: Write the simplest polynomial equation with roots -1, 2/3, and 4.
x = -1 or x = 2/3 or x = 4
x + 1 = 0 or 3x – 2 = 0 or x – 4 = 0
P(x) = (x + 1)(3x – 2)(x – 4)
P(x) = 3x3 - 11x2 - 6x + 8
White Board Activity:
Practice: Write the simplest polynomial function with the given zeros.
a. -2, 2, 4
b. 0, 2/3, 3
x = -2 or x = 2 or x = 4
x = 0 or x = 2/3, or x = 3
x + 2 = 0 or x – 2 = 0 or x – 4 = 0
x = 0 or 3x – 2 = 0 or x – 3 = 0
P(x) = (x + 2)(x – 2)(x – 4)
P(x) = x3 – 4x2 – 4x + 16
P(x) = x(3x – 2)(x – 3)
P(x) = 3x3 – 11x2 + 6x
Instruction:
Fundamental Theorem of Algebra
A polynomial of degree n, will have n zeros (including multiplicities).
These zeros may be real or complex.
If real, these zeros may be rational or irrational.
If irrational, they will occur as conjugate pairs.
Example: 3 + 5 and 3 - 5
If complex, they will occur as complex conjugate pairs.
Example: 3 + 5i and 3 – 5i
Open the book to page 191 and read example 3.
Example: Write the simplest polynomial function with zeros 1, 2i, 3 .
Complex zeros occur as conjugate pairs: 2i and -2i.
Irrational zeros occur as conjugate pairs: 3 , - 3 .
So x = 1 or x = 2i or x = -2i or x = 3 or x = - 3
So P(x) = (x – 1)(x – 2i)(x + 2i)(x – 3 )(x + 3 )
= (x – 1)(x2 – 4i2)(x2 – 3) = (x – 1)(x2 + 4)(x2 – 3)
= (x – 1)(x4 + x2 – 12)
P(x) = x5 – x4 + x3 – x2 – 12x + 12
White board Activity:
Practice: Write the simplest polynomial function with zeros 2, 3i, 2 .
Complex zeros occur as conjugate pairs: 3i, -3i.
Irrational zeros occur as conjugate pairs: 2 , - 2 .
So x = 2 or x = 3i or x = -3i or x = 2 or x = - 2
P(x) = (x – 2)(x – 3i)(x + 3i)(x - 2 )(x + 2 )
= (x – 2)(x2 – 9i2)(x2 – 2) = (x – 2)(x2 + 9)(x2 – 2)
= (x – 2)(x4 + 7x2 – 18)
P(x) = x5 – 2x4 +7x3 – 14x2 – 18x + 36
Open the book to page 190 and read example 2.
Example: Solve x4 – 3x3 + 5x2 - 27x – 36 = 0.
There will be 4 solutions including multiplicities.
Possible Rational Roots: ±1, ±2, ±4, ±6, ±9, ±12, ±18, ±36
Narrow the possibilities: -1, 4
Check -1:
Now check 4:
 1 | 1  3 5  27  36
4| 1 4
1
1 4
4
9
9  36
36
|0
1
9  36
4
0
0
9
36
|0
(x + 1)(x3 – 4x2 + 9x – 36) = 0
(x + 1)(x – 4)(x2 + 9) = 0
(x + 1)(x – 4)(x + 3i)(x – 3i) = 0
x = -1, 4, 3i, -3i
White Board Activity:
Practice: Solve x4 + 4x3 – x2 + 16x – 20 = 0 by finding all zeros.
There will be 4 solutions including multiplicities.
Possible rational roots: 1, -1, 2, -2, 4, -4, 5, -5, 10, -10, 20, -20
Narrow the possibilities: -5, 1
Check -5:
Now Check 1:
 5 | 1 4  1 16  20
1 | 1 1
5
1 1
5  20
4 4
20
|0
(x + 5)(x3 – x2 + 4x – 4 = 0
(x + 5)(x – 1)(x2 + 4) = 0
(x + 5)(x – 1)(x + 2i)(x – 2i) = 0
x = -5, 1, -2i, 2i
9 | 1 30
0  3159
9
351
1 39
351
3159
|0
Assessment:
Question student pairs.
Independent Practice:
Text: pg. 193 – 194 prob. 2 – 43.
For a Grade:
Text: pg. 193 – 194 prob. 2, 14, 20, 28.
1
4 4
1
0
4
0
4
|0