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
