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Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 1
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Chapter 2 Polynomial, Power, and Rational
Functions
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
2.6 Complex Zeros and the Fundamental
Theorem of Algebra
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 4
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± ±
= ± ±
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 5
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.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 6
Fundamental Theorem of Algebra
A polynomial function of degree n has n complex zeros (real and nonreal). Some of these zeros may be repeated.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 7
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
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 8
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).
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 9
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
= − +
= − − − +
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 10
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
= − + = += = −
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 11
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 12
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
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 13
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
= + − + − −
= + − − +
= − − + −
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 15
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 16
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.
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 17
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= + − − − −
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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
+ − − − −
= + − − = + − − = + − + = + − + +
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 19
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 2- 20
HW: p234, #1-41 odd
Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
2.7 Graphs of Rational Functions
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