Chapter 3. Polynomial and Rational Functions. 3.4 Zeros of Polynomial Functions

Сентябрь 14, 2022

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Chapter 3. Polynomial and Rational Functions. 3.4 Zeros of Polynomial Functions

Содержание

2. Use the Rational Zero Theorem to find possible rational zeros. Find zeros of a polynomial function.
3. The Rational Zero Theorem If has integer coefficients and (where is reduced to lowest terms) is
4. Example: Using the Rational Zero Theorem List all possible rational zeros of The constant term is
5. Example: Finding Zeros of a Polynomial Function Find all zeros of We begin by listing all
6. Example: Finding Zeros of a Polynomial Function (continued) Find all zeros of Possible rational zeros are
7. Example: Finding Zeros of a Polynomial Function (continued) Find all zeros of Possible rational zeros are
8. Example: Finding Zeros of a Polynomial Function (continued) Find all zeros of We have found a
9. Example: Finding Zeros of a Polynomial Function (continued) Find all zeros of We have found that
10. Properties of Roots of Polynomial Equations 1. If a polynomial equation is of degree n, then
11. Example: Solving a Polynomial Equation Solve We begin by listing all possible rational roots: Possible rational
12. Example: Solving a Polynomial Equation (continued) Solve Possible rational roots are 1, –1, 13, and –13.
13. Example: Solving a Polynomial Equation (continued) Solve We have found that x = 1 is a
14. Example: Solving a Polynomial Equation (continued) Solve Possible rational roots are 1, –1, 13, and –13.
15. Example: Solving a Polynomial Equation (continued) Solve The factored form of this polynomial is We will
16. The Fundamental Theorem of Algebra If f(x) is a polynomial of degree n, where then the
17. The Linear Factorization Theorem If where and then where c1, c2, ..., cn are complex numbers
18. Example: Finding a Polynomial Function with Given Zeros Find a third-degree polynomial function f(x) with real
19. Example: Finding a Polynomial Function with Given Zeros Find a third-degree polynomial function f(x) with real
20. Descartes’ Rule of Signs Let be a polynomial with real coefficients. 1. The number of positive
21. Descartes’ Rule of Signs (continued) Let be a polynomial with real coefficients. 2. The number of
22. Example: Using Descartes’ Rule of Signs Determine the possible number of positive and negative real zeros
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Слайд 2

Use the Rational Zero Theorem to find possible rational zeros.
Find zeros

of a polynomial function.
Solve polynomial equations.
Use the Linear Factorization Theorem to find polynomials with given zeros.
Use Descartes’ Rule of Signs.

Objectives:

Слайд 3

The Rational Zero Theorem
If has
integer coefficients and (where is reduced

to
lowest terms) is a rational zero of f, then p is a factor of the constant term, a0, and q is a factor of the leading coefficient, an.

Слайд 4

Example: Using the Rational Zero Theorem
List all possible rational zeros of
The

constant term is –3 and the leading coefficient is 4.
Factors of the constant term, –3:
Factors of the leading coefficient, 4:
Possible rational zeros are:

Слайд 5

Example: Finding Zeros of a Polynomial Function
Find all zeros of
We

begin by listing all possible rational zeros.
Possible rational zeros =
We now use synthetic division to see if we can find a rational zero among the four possible rational zeros.

Слайд 6

Example: Finding Zeros of a Polynomial Function (continued)
Find all zeros of

Possible rational zeros are 1, –1, 2, and –2. We will use synthetic division to test the possible rational zeros.
Neither –2 nor –1 is a zero. We continue testing possible rational zeros.

Слайд 7

Example: Finding Zeros of a Polynomial Function (continued)
Find all zeros of

Possible rational zeros are 1, –1, 2, and –2. We will use synthetic division to test the possible rational zeros. We have found that –2 and –1 are not rational zeros. We continue testing with 1 and 2.
We have found a rational zero at x = 2.

Слайд 8

Example: Finding Zeros of a Polynomial Function (continued)
Find all zeros of

We have found a rational zero at x = 2.
The result of synthetic division is:
This means that
We now solve

Слайд 9

Example: Finding Zeros of a Polynomial Function (continued)
Find all zeros of

We have found that
We now solve

The solution set is

The zeros of
are

Слайд 10

Properties of Roots of Polynomial Equations
1. If a polynomial equation is

of degree n, then counting multiple roots separately, the equation has n roots.
2. If a + bi is a root of a polynomial equation with real coefficients then the imaginary number
a – bi is also a root. Imaginary roots, if they exist, occur in conjugate pairs.

Слайд 11

Example: Solving a Polynomial Equation
Solve
We begin by listing all possible rational

roots:
Possible rational roots =
Possible rational roots are 1, –1, 13, and –13. We will use synthetic division to test the possible rational zeros.

Слайд 12

Example: Solving a Polynomial Equation (continued)
Solve
Possible rational roots are 1, –1,

13, and –13. We will use synthetic division to test the possible rational zeros.
x = 1 is a root for this polynomial.
We can rewrite the equation in factored form

Слайд 13

Example: Solving a Polynomial Equation (continued)
Solve
We have found that x =

1 is a root for this polynomial.
In factored form, the polynomial is
We now solve
We begin by listing all possible rational roots.
Possible rational roots =
Possible rational roots are 1, –1, 13, and –13. We will use synthetic division to test the possible rational zeros.

Слайд 14

Example: Solving a Polynomial Equation (continued)
Solve
Possible rational roots are 1, –1,

13, and –13. We will use synthetic division to test the possible rational zeros. Because –1 did not work for the original polynomial, it is not necessary to test that value.
The factored form of this polynomial is

x = 1 is a (repeated) root
for this polynomial

Слайд 15

Example: Solving a Polynomial Equation (continued)
Solve
The factored form of this polynomial

is
We will use the quadratic formula to solve

The solution set of the original
equation is

Слайд 16

The Fundamental Theorem of Algebra
If f(x) is a polynomial of degree

n, where then the equation f(x) = 0 has at least one complex root.

Слайд 17

The Linear Factorization Theorem
If where
and then
where c1, c2, ..., cn

are complex numbers (possibly real and not necessarily distinct). In words: An nth-degree polynomial can be expressed as the product of a nonzero constant and n linear factors, where each linear factor has a leading coefficient of 1.

Слайд 18

Example: Finding a Polynomial Function with Given Zeros
Find a third-degree polynomial

function f(x) with real coefficients that has –3 and i as zeros and such that
f(1) = 8.
Because i is a zero and the polynomial has real coefficients, the conjugate, –i, must also be a zero. We can now use the Linear Factorization Theorem.

Слайд 19

Example: Finding a Polynomial Function with Given Zeros
Find a third-degree polynomial

function f(x) with real coefficients that has –3 and i as zeros and such that
f(1) = 8.
Applying the Linear Factorization Theorem, we found
that

The polynomial function is

Слайд 20

Descartes’ Rule of Signs
Let
be a polynomial with real coefficients.
1. The number

of positive real zeros of f is either
a. the same as the number of sign changes of f(x)
or
b. less than the number of sign changes of f(x) by a
positive even integer. If f(x) has only one
variation in sign, then f has exactly one positive
real zero.

Слайд 21

Descartes’ Rule of Signs (continued)
Let
be a polynomial with real coefficients.
2. The

number of negative real zeros of f is either
a. The same as the number of sign changes in f(–x)
or
b. less than the number of sign changes in f(–x) by
a positive even integer. If f(–x) has only one
variation in sign, then f has exactly one negative
real zero

Слайд 22

Example: Using Descartes’ Rule of Signs
Determine the possible number of positive

and negative real zeros of
1. To find possibilities for positive real zeros, count the number of sign changes in the equation for f(x).
There are 4 variations in sign.
The number of positive real zeros of f is either 4, 2, or 0.

Chapter 3. Polynomial and Rational Functions. 3.4 Zeros of Polynomial Functions

Содержание

Use the Rational Zero Theorem to find possible rational zeros.Find zeros

The Rational Zero TheoremIf has integer coefficients and (where is reduced

Example: Using the Rational Zero TheoremList all possible rational zeros ofThe

Example: Finding Zeros of a Polynomial FunctionFind all zeros of We

Example: Finding Zeros of a Polynomial Function (continued)Find all zeros of

Example: Finding Zeros of a Polynomial Function (continued)Find all zeros of

Example: Finding Zeros of a Polynomial Function (continued)Find all zeros of

Example: Finding Zeros of a Polynomial Function (continued)Find all zeros of

Properties of Roots of Polynomial Equations1. If a polynomial equation is

Example: Solving a Polynomial EquationSolveWe begin by listing all possible rational

Example: Solving a Polynomial Equation (continued)SolvePossible rational roots are 1, –1,

Example: Solving a Polynomial Equation (continued)SolveWe have found that x =

Example: Solving a Polynomial Equation (continued)SolvePossible rational roots are 1, –1,

Example: Solving a Polynomial Equation (continued)SolveThe factored form of this polynomial

The Fundamental Theorem of AlgebraIf f(x) is a polynomial of degree

The Linear Factorization TheoremIf where and thenwhere c1, c2, ..., cn

Example: Finding a Polynomial Function with Given ZerosFind a third-degree polynomial

Example: Finding a Polynomial Function with Given ZerosFind a third-degree polynomial

Descartes’ Rule of SignsLetbe a polynomial with real coefficients.1. The number

Descartes’ Rule of Signs (continued)Letbe a polynomial with real coefficients.2. The

Example: Using Descartes’ Rule of SignsDetermine the possible number of positive

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