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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Chapter 2 Polynomial, Power, and Rational

Functions

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2.6 Complex Zeros and the Fundamental

Theorem of Algebra

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Quick Review Solutions

( ) ( )

( )( )

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 potentia

a bi

i i

i i

x x

x x

+

+ + − +

− + −

− −

− + =

4 2

l rational zeros.

5. 4 3 2x x x− + +

1 8i= +

1 18i= − +

( )( )2 1 5x x= + −

3x i= = ±

1 12, 1, ,2 4± ±

= ± ±

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What you’ll learn about Two Major Theorems Complex Conjugate Zeros Factoring with Real Number Coefficients … and why These topics provide the complete story about the zeros and factors of polynomials with real number coefficients.

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Fundamental Theorem of Algebra

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

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Linear Factorization Theorem

1 2

1 2

If ( ) is a polynomial function of degree 0, then ( ) has precisely linear factors and ( ) ( )( )...( ) where is the leading coefficient of ( ) and , ,..., are the complex ze

n

n

f x n f xn f x a x z x z x z a

f x z z z

>= − − −

ros of ( ).The are not necessarily distinct numbers; some may be repeated.i

f xz

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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) = 0 2. k is a zero of the function f. 3. x – k is a factor of f(x).

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Example Exploring Fundamental Polynomial Connections

( ) ( )( )

Write the polynomial function in standard form, identify the zeros of the function and the -intercepts of its graph.

) ( ) ( 2 )( 2 )

) 3 3 ( )( )

xa f x x i x i

b f x x x x i x i

= − +

= − − − +

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Example Exploring Fundamental Polynomial Connections

2The function ( ) ( 2 )( 2 ) 4 has two zeros:

2 and 2 . Because the zeros are not real, the graph of has no -intercepts.

f x x i x i xx i x i

f x

= − + = += = −

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Complex Conjugate Zeros

Suppose that ( ) is a polynomial function with coefficients. If and arereal numbers with 0 and is a zero of ( ), then its complex conjugate

is also a zero of ( ).

f x a bb a bi f x

a bi f x≠ +

−

real

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

x xi i

x i x i

− − ++ −

− + − −

( )( )2 2

4 3 2

( ) - 2 3 [ (1 )][ (1 )]

( 6)( 2 2)

6 14 12

f x x x x i x i

x x x xx x x x

= + − + − −

= + − − +

= − − + −

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Finding Complex Zeros

Slide 2- 14

4 2The complex number z = 1-2i is a zero of f (x)= 4x +17x +14x+65. Findthe remaining zeros of f x, and write it in its linear factorization.

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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.

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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= + − − − −

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Slide 2- 18

Example Factoring a Polynomial The Rational Zeros Theorem provides the candidates for the rationalzeros of . The graph of suggests which candidates to try first.Using synthetic division, find that 1/ 3 is a zero. Thus,

( )

f fx

f x

= −

=

( )

( )( )

( )( )

( )( )( )

5 4 3 2

4 2

4 2

2 2

2

3 24 8 27 91 3 24 2731 3 8 93

1 3 9 13

1 3 3 3 13

x x x x x

x x x

x x x

x x x

x x x x

+ − − − −

= + − − = + − − = + − + = + − + +

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HW: p234, #1-41 odd

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2.7 Graphs of Rational Functions