section 3.4 zeros of polynomial functions. the rational zero theorem
TRANSCRIPT
![Page 1: Section 3.4 Zeros of Polynomial Functions. The Rational Zero Theorem](https://reader033.vdocuments.net/reader033/viewer/2022061418/56649da65503460f94a91e13/html5/thumbnails/1.jpg)
Section 3.4Zeros of Polynomial Functions
![Page 2: Section 3.4 Zeros of Polynomial Functions. The Rational Zero Theorem](https://reader033.vdocuments.net/reader033/viewer/2022061418/56649da65503460f94a91e13/html5/thumbnails/2.jpg)
The Rational Zero Theorem
![Page 3: Section 3.4 Zeros of Polynomial Functions. The Rational Zero Theorem](https://reader033.vdocuments.net/reader033/viewer/2022061418/56649da65503460f94a91e13/html5/thumbnails/3.jpg)
![Page 4: Section 3.4 Zeros of Polynomial Functions. The Rational Zero Theorem](https://reader033.vdocuments.net/reader033/viewer/2022061418/56649da65503460f94a91e13/html5/thumbnails/4.jpg)
Example
List all possible rational zeros of f(x)=x3-3x2-4x+12
Find one of the zeros of the function using synthetic division, then factor the remaining polynomial. What are all of the zeros of the function? How can the graph below help you find the zeros?
![Page 5: Section 3.4 Zeros of Polynomial Functions. The Rational Zero Theorem](https://reader033.vdocuments.net/reader033/viewer/2022061418/56649da65503460f94a91e13/html5/thumbnails/5.jpg)
Example
List all possible rational zeros of f(x)=6x3-19x2+2x+3
Starting with the integers, find one zero of the function using synthetic division, then factor the remaining polynomial. What are all of the zeros of the function?
![Page 6: Section 3.4 Zeros of Polynomial Functions. The Rational Zero Theorem](https://reader033.vdocuments.net/reader033/viewer/2022061418/56649da65503460f94a91e13/html5/thumbnails/6.jpg)
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 those roots. Factor the remaining polynomial. What are all of the roots of the equation? The graph below will NOT help you find the imaginary roots. Why?
![Page 7: Section 3.4 Zeros of Polynomial Functions. The Rational Zero Theorem](https://reader033.vdocuments.net/reader033/viewer/2022061418/56649da65503460f94a91e13/html5/thumbnails/7.jpg)
![Page 8: Section 3.4 Zeros of Polynomial Functions. The Rational Zero Theorem](https://reader033.vdocuments.net/reader033/viewer/2022061418/56649da65503460f94a91e13/html5/thumbnails/8.jpg)
4 3
Notice that the roots for our most recent problem
(x -x 7 9 18 0; degree 4) were 3i,2,-1x x
![Page 9: Section 3.4 Zeros of Polynomial Functions. The Rational Zero Theorem](https://reader033.vdocuments.net/reader033/viewer/2022061418/56649da65503460f94a91e13/html5/thumbnails/9.jpg)
The Fundamental Theorem of Algebra
![Page 10: Section 3.4 Zeros of Polynomial Functions. The Rational Zero Theorem](https://reader033.vdocuments.net/reader033/viewer/2022061418/56649da65503460f94a91e13/html5/thumbnails/10.jpg)
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.
![Page 11: Section 3.4 Zeros of Polynomial Functions. The Rational Zero Theorem](https://reader033.vdocuments.net/reader033/viewer/2022061418/56649da65503460f94a91e13/html5/thumbnails/11.jpg)
The Linear Factorization Theorem
![Page 12: Section 3.4 Zeros of Polynomial Functions. The Rational Zero Theorem](https://reader033.vdocuments.net/reader033/viewer/2022061418/56649da65503460f94a91e13/html5/thumbnails/12.jpg)
![Page 13: Section 3.4 Zeros of Polynomial Functions. The Rational Zero Theorem](https://reader033.vdocuments.net/reader033/viewer/2022061418/56649da65503460f94a91e13/html5/thumbnails/13.jpg)
Example
Find a fourth-degree polynomial function f(x) with real coefficients that has -1,2 and i as zeros and such that f(1)=- 4
![Page 14: Section 3.4 Zeros of Polynomial Functions. The Rational Zero Theorem](https://reader033.vdocuments.net/reader033/viewer/2022061418/56649da65503460f94a91e13/html5/thumbnails/14.jpg)
Descartes’s Rule of Signs
![Page 15: Section 3.4 Zeros of Polynomial Functions. The Rational Zero Theorem](https://reader033.vdocuments.net/reader033/viewer/2022061418/56649da65503460f94a91e13/html5/thumbnails/15.jpg)
![Page 16: Section 3.4 Zeros of Polynomial Functions. The Rational Zero Theorem](https://reader033.vdocuments.net/reader033/viewer/2022061418/56649da65503460f94a91e13/html5/thumbnails/16.jpg)
![Page 17: Section 3.4 Zeros of Polynomial Functions. The Rational Zero Theorem](https://reader033.vdocuments.net/reader033/viewer/2022061418/56649da65503460f94a91e13/html5/thumbnails/17.jpg)
Descartes’s 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.
![Page 18: Section 3.4 Zeros of Polynomial Functions. The Rational Zero Theorem](https://reader033.vdocuments.net/reader033/viewer/2022061418/56649da65503460f94a91e13/html5/thumbnails/18.jpg)
Example
For f(x)=x3- 3x2- x+3 how many positive and negative zeros are there? What are the zeros of the function?
![Page 19: Section 3.4 Zeros of Polynomial Functions. The Rational Zero Theorem](https://reader033.vdocuments.net/reader033/viewer/2022061418/56649da65503460f94a91e13/html5/thumbnails/19.jpg)
Example
For f(x)=x3- x2+4x- 4 how many positive and negative zeros are there? Use a graphing utility to find one real zero of the function. What are all the zeros of the function?
![Page 20: Section 3.4 Zeros of Polynomial Functions. The Rational Zero Theorem](https://reader033.vdocuments.net/reader033/viewer/2022061418/56649da65503460f94a91e13/html5/thumbnails/20.jpg)
(a)
(b)
(c)
(d)
List all possible rational zeros of the function f(x)=x3+3x2- 6x-8.
1, 1, 2, 4, 8
21, 2, 4, 8
1, 2, 4,
21,
![Page 21: Section 3.4 Zeros of Polynomial Functions. The Rational Zero Theorem](https://reader033.vdocuments.net/reader033/viewer/2022061418/56649da65503460f94a91e13/html5/thumbnails/21.jpg)
(a)
(b)
(c)
(d)
Find a third-degree polynomial function f(x) with real coefficients that have 1 and 2i as zeros and such that f(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
![Page 22: Section 3.4 Zeros of Polynomial Functions. The Rational Zero Theorem](https://reader033.vdocuments.net/reader033/viewer/2022061418/56649da65503460f94a91e13/html5/thumbnails/22.jpg)
(a)
(b)
(c)
(d)
What are the zeros of the function f(x)=x3+2x2+8x+16? Find the first zero using a graphing utility.
2, 2 2,2 2
2, 2 2,2 2
2, 2 ,2
2, 2,2
i i
i i