Section 2.5 Zeros of Polynomial Functions of Polynomial... · Section 2.5 Zeros of Polynomial...
Transcript of Section 2.5 Zeros of Polynomial Functions of Polynomial... · Section 2.5 Zeros of Polynomial...
![Page 1: Section 2.5 Zeros of Polynomial Functions of Polynomial... · Section 2.5 Zeros of Polynomial Functions The Rational Zero Theorem Example List all possible rational zeros of f(x)=x3-3x2-4x+12](https://reader036.fdocuments.net/reader036/viewer/2022070107/6022a8c8051dda10c17ee277/html5/thumbnails/1.jpg)
Section 2.5
Zeros of Polynomial Functions
![Page 2: Section 2.5 Zeros of Polynomial Functions of Polynomial... · Section 2.5 Zeros of Polynomial Functions The Rational Zero Theorem Example List all possible rational zeros of f(x)=x3-3x2-4x+12](https://reader036.fdocuments.net/reader036/viewer/2022070107/6022a8c8051dda10c17ee277/html5/thumbnails/2.jpg)
The Rational Zero Theorem
![Page 3: Section 2.5 Zeros of Polynomial Functions of Polynomial... · Section 2.5 Zeros of Polynomial Functions The Rational Zero Theorem Example List all possible rational zeros of f(x)=x3-3x2-4x+12](https://reader036.fdocuments.net/reader036/viewer/2022070107/6022a8c8051dda10c17ee277/html5/thumbnails/3.jpg)
![Page 4: Section 2.5 Zeros of Polynomial Functions of Polynomial... · Section 2.5 Zeros of Polynomial Functions The Rational Zero Theorem Example List all possible rational zeros of f(x)=x3-3x2-4x+12](https://reader036.fdocuments.net/reader036/viewer/2022070107/6022a8c8051dda10c17ee277/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?
![Page 5: Section 2.5 Zeros of Polynomial Functions of Polynomial... · Section 2.5 Zeros of Polynomial Functions The Rational Zero Theorem Example List all possible rational zeros of f(x)=x3-3x2-4x+12](https://reader036.fdocuments.net/reader036/viewer/2022070107/6022a8c8051dda10c17ee277/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 2.5 Zeros of Polynomial Functions of Polynomial... · Section 2.5 Zeros of Polynomial Functions The Rational Zero Theorem Example List all possible rational zeros of f(x)=x3-3x2-4x+12](https://reader036.fdocuments.net/reader036/viewer/2022070107/6022a8c8051dda10c17ee277/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. Factor the remaining polynomial. What are all of
the roots of the equation?
![Page 7: Section 2.5 Zeros of Polynomial Functions of Polynomial... · Section 2.5 Zeros of Polynomial Functions The Rational Zero Theorem Example List all possible rational zeros of f(x)=x3-3x2-4x+12](https://reader036.fdocuments.net/reader036/viewer/2022070107/6022a8c8051dda10c17ee277/html5/thumbnails/7.jpg)
![Page 8: Section 2.5 Zeros of Polynomial Functions of Polynomial... · Section 2.5 Zeros of Polynomial Functions The Rational Zero Theorem Example List all possible rational zeros of f(x)=x3-3x2-4x+12](https://reader036.fdocuments.net/reader036/viewer/2022070107/6022a8c8051dda10c17ee277/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 2.5 Zeros of Polynomial Functions of Polynomial... · Section 2.5 Zeros of Polynomial Functions The Rational Zero Theorem Example List all possible rational zeros of f(x)=x3-3x2-4x+12](https://reader036.fdocuments.net/reader036/viewer/2022070107/6022a8c8051dda10c17ee277/html5/thumbnails/9.jpg)
The Fundamental Theorem of
Algebra
![Page 10: Section 2.5 Zeros of Polynomial Functions of Polynomial... · Section 2.5 Zeros of Polynomial Functions The Rational Zero Theorem Example List all possible rational zeros of f(x)=x3-3x2-4x+12](https://reader036.fdocuments.net/reader036/viewer/2022070107/6022a8c8051dda10c17ee277/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 2.5 Zeros of Polynomial Functions of Polynomial... · Section 2.5 Zeros of Polynomial Functions The Rational Zero Theorem Example List all possible rational zeros of f(x)=x3-3x2-4x+12](https://reader036.fdocuments.net/reader036/viewer/2022070107/6022a8c8051dda10c17ee277/html5/thumbnails/11.jpg)
The Linear Factorization
Theorem
![Page 12: Section 2.5 Zeros of Polynomial Functions of Polynomial... · Section 2.5 Zeros of Polynomial Functions The Rational Zero Theorem Example List all possible rational zeros of f(x)=x3-3x2-4x+12](https://reader036.fdocuments.net/reader036/viewer/2022070107/6022a8c8051dda10c17ee277/html5/thumbnails/12.jpg)
![Page 13: Section 2.5 Zeros of Polynomial Functions of Polynomial... · Section 2.5 Zeros of Polynomial Functions The Rational Zero Theorem Example List all possible rational zeros of f(x)=x3-3x2-4x+12](https://reader036.fdocuments.net/reader036/viewer/2022070107/6022a8c8051dda10c17ee277/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 2.5 Zeros of Polynomial Functions of Polynomial... · Section 2.5 Zeros of Polynomial Functions The Rational Zero Theorem Example List all possible rational zeros of f(x)=x3-3x2-4x+12](https://reader036.fdocuments.net/reader036/viewer/2022070107/6022a8c8051dda10c17ee277/html5/thumbnails/14.jpg)
Descartes’s Rule of Signs
![Page 15: Section 2.5 Zeros of Polynomial Functions of Polynomial... · Section 2.5 Zeros of Polynomial Functions The Rational Zero Theorem Example List all possible rational zeros of f(x)=x3-3x2-4x+12](https://reader036.fdocuments.net/reader036/viewer/2022070107/6022a8c8051dda10c17ee277/html5/thumbnails/15.jpg)
![Page 16: Section 2.5 Zeros of Polynomial Functions of Polynomial... · Section 2.5 Zeros of Polynomial Functions The Rational Zero Theorem Example List all possible rational zeros of f(x)=x3-3x2-4x+12](https://reader036.fdocuments.net/reader036/viewer/2022070107/6022a8c8051dda10c17ee277/html5/thumbnails/16.jpg)
![Page 17: Section 2.5 Zeros of Polynomial Functions of Polynomial... · Section 2.5 Zeros of Polynomial Functions The Rational Zero Theorem Example List all possible rational zeros of f(x)=x3-3x2-4x+12](https://reader036.fdocuments.net/reader036/viewer/2022070107/6022a8c8051dda10c17ee277/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 2.5 Zeros of Polynomial Functions of Polynomial... · Section 2.5 Zeros of Polynomial Functions The Rational Zero Theorem Example List all possible rational zeros of f(x)=x3-3x2-4x+12](https://reader036.fdocuments.net/reader036/viewer/2022070107/6022a8c8051dda10c17ee277/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 2.5 Zeros of Polynomial Functions of Polynomial... · Section 2.5 Zeros of Polynomial Functions The Rational Zero Theorem Example List all possible rational zeros of f(x)=x3-3x2-4x+12](https://reader036.fdocuments.net/reader036/viewer/2022070107/6022a8c8051dda10c17ee277/html5/thumbnails/19.jpg)
Example
For f(x)=x3- x2+4x- 4 how many positive and
negative zeros are there? What are all the
zeros of the function?
![Page 20: Section 2.5 Zeros of Polynomial Functions of Polynomial... · Section 2.5 Zeros of Polynomial Functions The Rational Zero Theorem Example List all possible rational zeros of f(x)=x3-3x2-4x+12](https://reader036.fdocuments.net/reader036/viewer/2022070107/6022a8c8051dda10c17ee277/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
2
1, 2, 4, 8
1, 2, 4,
2
1,
![Page 21: Section 2.5 Zeros of Polynomial Functions of Polynomial... · Section 2.5 Zeros of Polynomial Functions The Rational Zero Theorem Example List all possible rational zeros of f(x)=x3-3x2-4x+12](https://reader036.fdocuments.net/reader036/viewer/2022070107/6022a8c8051dda10c17ee277/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 2.5 Zeros of Polynomial Functions of Polynomial... · Section 2.5 Zeros of Polynomial Functions The Rational Zero Theorem Example List all possible rational zeros of f(x)=x3-3x2-4x+12](https://reader036.fdocuments.net/reader036/viewer/2022070107/6022a8c8051dda10c17ee277/html5/thumbnails/22.jpg)
(a)
(b)
(c)
(d)
What are the zeros of the function
f(x)=x3+2x2+8x+16?
2, 2 2,2 2
2, 2 2,2 2
2, 2 ,2
2, 2,2
i i
i i