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Section 2.5Zeros of Polynomial Functions

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The Rational Zero Theorem

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Example

List all possible rational zeros of f(x)=x3

-3x2

-4x+12Find one of the zeros of the function using synthetic division, thenfactor the remaining polynomial. What are all of the zeros of thefunction? How can the graph below help you find the zeros?

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Example

List all possible rational zeros of f(x)=6x3

-19x2

+2x+3Starting with the integers, find one zero of the function usingsynthetic division, then factor the remaining polynomial. What areall of the zeros of the function?

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Example

List all possible rational roots of x4-x3+7x2- 9x-18=0

Starting with the integers, find two roots of the equation using

synthetic division. The graph below will help you easily find thoseroots. Factor the remaining polynomial. What are all of the roots ofthe equation? The graph below will NOT help you find the

imaginary roots. Why?

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4 3

Notice that the roots for our most recent problem

(x -x 7 9 18 0; degree 4) were 3i,2,-1x x

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The Fundamental Theorem ofAlgebra

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Remember that having roots of 3, -2, etc. are

complex roots because 3 can be written 3+0i

and -2 can be written as -2+0i.

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The Linear FactorizationTheorem

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Example

Find a fourth-degree polynomial functionf(x) with real coefficients that has -1,2 and i

as zeros and such that f(1)=- 4

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Descartess Rule of Signs

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Descartess Rule of Signs

3 2

Determine the possible numbers of positive and negative

real zeros of f(x)=x 2 5 6.

To find possibilities for positive real zeros, count the number

of sign changes in the equation for f(x). There

x x

23

3 2

is one variation

in sign change, so there is one positive real zero.

Now substitute in -x for x:

f(x)=(-x) 2 5( ) 6

f(x)= - x 2 5 6

There are two sign changes so there are either 2 negative

real z

x x

x x

eros or none. There has to be 2 to give you a total of 3.

The zeros are 2, -1, -3.

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Example

For f(x)=x3

- 3x2

- x+3 how many positive andnegative zeros are there? What are the

zeros of the function?

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Example

For f(x)=x3- x2+4x- 4 how many positive andnegative zeros are there? Use a graphingutility to find one real zero of the function.What are all the zeros of the function?

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(a)

(b)

(c)

(d)

List all possible rational zeros of thefunction f(x)=x3+3x2- 6x-8.

1, 1, 2, 4, 8

2

1, 2, 4, 8

1, 2, 4,

2

1,

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(a)

(b)

(c)

(d)

Find a third-degree polynomialfunction f(x) with real coefficients that

have 1 and 2ias zeros and such thatf(1)=0.

3 2

3 2

3 2

3 2

( ) 4 4

( ) 2 4 8

( ) 2 2 8 8

( ) 4 4

f x x x x

f x x x x

f x x x x

f x x x x

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(a)

(b)

(c)

(d)

What are the zeros of the function

f(x)=x3+2x2+8x+16? Find the first zerousing a graphing utility.

2, 2 2,2 2

2, 2 2,2 2

2, 2 ,2

2, 2,2

i i

i i