2.12.12.12.1

Complex Zeros and the Complex Zeros and the Fundamental Theorem of Fundamental Theorem of

AlgebraAlgebra

Quick Review

2

2

Perform the indicated operation, and write the result in the form .

1. 2 3 1 5

2. 3 2 3 4

Factor the quadratic equation.

3. 2 9 5

Solve the quadratic equation.

4. 6 10 0

List all potential ra

a bi

i i

i i

x x

x x

4 2

tional zeros.

5. 4 3 2x x x

Quick Review Solutions

2

1

Perform the indicated operation, and write the result in the form .

1. 2 3 1 5

2. 3 2 3 4

Factor the quadratic equation.

3. 2 9 5

Solve the quadratic equatio

8

1 18

2 1 5

n.

a bi

i i

i i

i

i

x xx x

2

4 2

4. 6 10 0

List all potential rational zeros.

3

2, 1, 15. 4 3 2 / 2, 1/ 4

x x

x x

x i

x

What you’ll learn about• Complex Numbers• Two Major Theorems• Complex Conjugate Zeros• Factoring with Real Number Coefficients

… and whyThese topics provide the complete story about the zeros and factors of polynomials with real

number coefficients.

Fundamental Theorem of Algebra

A polynomial function of degree n has n complex zeros (real and nonreal). Some of these zeros may be repeated.

Fundamental Polynomial Connections in the Complex

Case

The following statements about a polynomial function f are equivalent if k is a complex number:

1. x = k is a solution (or root) of the equation f(x) = 02. k is a zero of the function f.

3. x – k is a factor of f(x).

Example Exploring Fundamental Polynomial

ConnectionsWrite the polynomial function in standard form, identify the zeros of the function

and the -intercepts of its graph.

( ) ( 3 )( 3 )

x

f x x i x i

Example Exploring Fundamental Polynomial

Connections

Write the polynomial function in standard form, identify the zeros of the function

and the -intercepts of its graph.

( ) ( 3 )( 3 )

x

f x x i x i

2The function ( ) ( 3 )( 3 ) 9 has two zeros:

3 and 3 . Because the zeros are not real, the

graph of has no -intercepts.

f x x i x i x

x i x i

f x

Complex Conjugate Zeros

Suppose that ( ) is a polynomial function with real coefficients. If and are

real numbers with 0 and is a zero of ( ), then its complex conjugate

is also a zero of ( ).

f x a b

b a bi f x

a bi f x

Example Finding a Polynomial from Given

ZerosWrite a polynomial of minimum degree in standard form with real

coefficients whose zeros include 2, 3, and 1 .i

Example Finding a Polynomial from Given

Zeros

Write a polynomial of minimum degree in standard form with real

coefficients whose zeros include 2, 3, and 1 .i

Because 2 and 3 are real zeros, 2 and 3 must be factors.

Because the coefficients are real and 1 is a zero, 1 must also

be a zero. Therefore, (1 ) and (1 ) must be factors.

( ) - 2 3 [

x x

i i

x i x i

f x x x x

2 2

4 3 2

(1 )][ (1 )]

( 6)( 2 2)

6 14 12

i x i

x x x x

x x x x

Factors of a Polynomial with Real Coefficients

Every polynomial function with real coefficients can be written as a product of linear factors and irreducible quadratic factors, each with real coefficients.

Example Factoring a Polynomial

5 4 3 2Write ( ) 3 24 8 27 9as a product of linear and

irreducible quadratic factors, each with real coefficients.

f x x x x x x

Example Factoring a Polynomial

5 4 3 2Write ( ) 3 24 8 27 9as a product of linear and

irreducible quadratic factors, each with real coefficients.

f x x x x x x

The Rational Zeros Theorem provides the candidates for the rational

zeros of . The graph of suggests which candidates to try first.

Using synthetic division, find that 1/3 is a zero. Thus,

( ) 3

f f

x

f x x

5 4 3 2

4 2

2 2

2

24 8 27 9

1 3 8 9

3

1 3 9 1

3

1 3 3 3 1

3

x x x x

x x x

x x x

x x x x