2.1 2.1 Complex Zeros and the Complex Zeros and the Fundamental Theorem of Fundamental Theorem of Algebra Algebra
Jan 19, 2016
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