3.3 graphs of polynomial functions - dr. travers page of...
TRANSCRIPT
![Page 1: 3.3 Graphs of Polynomial Functions - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/graphs_of_polynomial_function… · How can we find the roots of a polynomial?](https://reader035.vdocuments.mx/reader035/viewer/2022071212/602443539f43ac2b6a4c4185/html5/thumbnails/1.jpg)
3.3 Graphs of Polynomial Functions
![Page 2: 3.3 Graphs of Polynomial Functions - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/graphs_of_polynomial_function… · How can we find the roots of a polynomial?](https://reader035.vdocuments.mx/reader035/viewer/2022071212/602443539f43ac2b6a4c4185/html5/thumbnails/2.jpg)
Roots
In this section, we want to look at how to take a function andbreak it into the pieces we need to be able to sketch the graph.
DefinitionA root of a polynomial f is a value x such that f (x) = 0
Does anyone know the Fundamental Theorem of Algebra? Forour purposes, it says ...
TheoremFundamental Theorem of AlgebraEvery real values polynomial of degree n has n roots that are notnecessarily distinct.
![Page 3: 3.3 Graphs of Polynomial Functions - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/graphs_of_polynomial_function… · How can we find the roots of a polynomial?](https://reader035.vdocuments.mx/reader035/viewer/2022071212/602443539f43ac2b6a4c4185/html5/thumbnails/3.jpg)
Roots
In this section, we want to look at how to take a function andbreak it into the pieces we need to be able to sketch the graph.
DefinitionA root of a polynomial f is a value x such that f (x) = 0
Does anyone know the Fundamental Theorem of Algebra? Forour purposes, it says ...
TheoremFundamental Theorem of AlgebraEvery real values polynomial of degree n has n roots that are notnecessarily distinct.
![Page 4: 3.3 Graphs of Polynomial Functions - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/graphs_of_polynomial_function… · How can we find the roots of a polynomial?](https://reader035.vdocuments.mx/reader035/viewer/2022071212/602443539f43ac2b6a4c4185/html5/thumbnails/4.jpg)
Roots
In this section, we want to look at how to take a function andbreak it into the pieces we need to be able to sketch the graph.
DefinitionA root of a polynomial f is a value x such that f (x) = 0
Does anyone know the Fundamental Theorem of Algebra?
Forour purposes, it says ...
TheoremFundamental Theorem of AlgebraEvery real values polynomial of degree n has n roots that are notnecessarily distinct.
![Page 5: 3.3 Graphs of Polynomial Functions - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/graphs_of_polynomial_function… · How can we find the roots of a polynomial?](https://reader035.vdocuments.mx/reader035/viewer/2022071212/602443539f43ac2b6a4c4185/html5/thumbnails/5.jpg)
Roots
In this section, we want to look at how to take a function andbreak it into the pieces we need to be able to sketch the graph.
DefinitionA root of a polynomial f is a value x such that f (x) = 0
Does anyone know the Fundamental Theorem of Algebra? Forour purposes, it says ...
TheoremFundamental Theorem of AlgebraEvery real values polynomial of degree n has n roots that are notnecessarily distinct.
![Page 6: 3.3 Graphs of Polynomial Functions - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/graphs_of_polynomial_function… · How can we find the roots of a polynomial?](https://reader035.vdocuments.mx/reader035/viewer/2022071212/602443539f43ac2b6a4c4185/html5/thumbnails/6.jpg)
Roots
In this section, we want to look at how to take a function andbreak it into the pieces we need to be able to sketch the graph.
DefinitionA root of a polynomial f is a value x such that f (x) = 0
Does anyone know the Fundamental Theorem of Algebra? Forour purposes, it says ...
TheoremFundamental Theorem of AlgebraEvery real values polynomial of degree n has n roots that are notnecessarily distinct.
![Page 7: 3.3 Graphs of Polynomial Functions - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/graphs_of_polynomial_function… · How can we find the roots of a polynomial?](https://reader035.vdocuments.mx/reader035/viewer/2022071212/602443539f43ac2b6a4c4185/html5/thumbnails/7.jpg)
Finding Roots
How can we find the roots of a polynomial?
Factoring
If we can write a polynomial in factored form, we can find theroots of the polynomial.
Example
Find the roots of f (x) = x2 − 4x + 3
x2 − 4x + 3 = 0(x− 3)(x− 1) = 0
x = 1, 3
![Page 8: 3.3 Graphs of Polynomial Functions - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/graphs_of_polynomial_function… · How can we find the roots of a polynomial?](https://reader035.vdocuments.mx/reader035/viewer/2022071212/602443539f43ac2b6a4c4185/html5/thumbnails/8.jpg)
Finding Roots
How can we find the roots of a polynomial?
Factoring
If we can write a polynomial in factored form, we can find theroots of the polynomial.
Example
Find the roots of f (x) = x2 − 4x + 3
x2 − 4x + 3 = 0(x− 3)(x− 1) = 0
x = 1, 3
![Page 9: 3.3 Graphs of Polynomial Functions - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/graphs_of_polynomial_function… · How can we find the roots of a polynomial?](https://reader035.vdocuments.mx/reader035/viewer/2022071212/602443539f43ac2b6a4c4185/html5/thumbnails/9.jpg)
Finding Roots
How can we find the roots of a polynomial?
Factoring
If we can write a polynomial in factored form, we can find theroots of the polynomial.
Example
Find the roots of f (x) = x2 − 4x + 3
x2 − 4x + 3 = 0(x− 3)(x− 1) = 0
x = 1, 3
![Page 10: 3.3 Graphs of Polynomial Functions - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/graphs_of_polynomial_function… · How can we find the roots of a polynomial?](https://reader035.vdocuments.mx/reader035/viewer/2022071212/602443539f43ac2b6a4c4185/html5/thumbnails/10.jpg)
Finding Roots
How can we find the roots of a polynomial?
Factoring
If we can write a polynomial in factored form, we can find theroots of the polynomial.
Example
Find the roots of f (x) = x2 − 4x + 3
x2 − 4x + 3 = 0
(x− 3)(x− 1) = 0x = 1, 3
![Page 11: 3.3 Graphs of Polynomial Functions - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/graphs_of_polynomial_function… · How can we find the roots of a polynomial?](https://reader035.vdocuments.mx/reader035/viewer/2022071212/602443539f43ac2b6a4c4185/html5/thumbnails/11.jpg)
Finding Roots
How can we find the roots of a polynomial?
Factoring
If we can write a polynomial in factored form, we can find theroots of the polynomial.
Example
Find the roots of f (x) = x2 − 4x + 3
x2 − 4x + 3 = 0(x− 3)(x− 1) = 0
x = 1, 3
![Page 12: 3.3 Graphs of Polynomial Functions - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/graphs_of_polynomial_function… · How can we find the roots of a polynomial?](https://reader035.vdocuments.mx/reader035/viewer/2022071212/602443539f43ac2b6a4c4185/html5/thumbnails/12.jpg)
Finding Roots
How can we find the roots of a polynomial?
Factoring
If we can write a polynomial in factored form, we can find theroots of the polynomial.
Example
Find the roots of f (x) = x2 − 4x + 3
x2 − 4x + 3 = 0(x− 3)(x− 1) = 0
x = 1, 3
![Page 13: 3.3 Graphs of Polynomial Functions - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/graphs_of_polynomial_function… · How can we find the roots of a polynomial?](https://reader035.vdocuments.mx/reader035/viewer/2022071212/602443539f43ac2b6a4c4185/html5/thumbnails/13.jpg)
Finding Roots
Example
Find the roots of f (x) = x3 − 3x2 − 4x + 12.
x3 − 3x2 − 4x + 12 = 0x2(x− 3)− 4(x− 3) = 0
(x− 3)(x2 − 4) = 0(x− 3)(x− 2)(x + 2) = 0
x = ±2, 3
![Page 14: 3.3 Graphs of Polynomial Functions - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/graphs_of_polynomial_function… · How can we find the roots of a polynomial?](https://reader035.vdocuments.mx/reader035/viewer/2022071212/602443539f43ac2b6a4c4185/html5/thumbnails/14.jpg)
Finding Roots
Example
Find the roots of f (x) = x3 − 3x2 − 4x + 12.
x3 − 3x2 − 4x + 12 = 0
x2(x− 3)− 4(x− 3) = 0(x− 3)(x2 − 4) = 0
(x− 3)(x− 2)(x + 2) = 0x = ±2, 3
![Page 15: 3.3 Graphs of Polynomial Functions - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/graphs_of_polynomial_function… · How can we find the roots of a polynomial?](https://reader035.vdocuments.mx/reader035/viewer/2022071212/602443539f43ac2b6a4c4185/html5/thumbnails/15.jpg)
Finding Roots
Example
Find the roots of f (x) = x3 − 3x2 − 4x + 12.
x3 − 3x2 − 4x + 12 = 0x2(x− 3)− 4(x− 3) = 0
(x− 3)(x2 − 4) = 0(x− 3)(x− 2)(x + 2) = 0
x = ±2, 3
![Page 16: 3.3 Graphs of Polynomial Functions - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/graphs_of_polynomial_function… · How can we find the roots of a polynomial?](https://reader035.vdocuments.mx/reader035/viewer/2022071212/602443539f43ac2b6a4c4185/html5/thumbnails/16.jpg)
Finding Roots
Example
Find the roots of f (x) = x3 − 3x2 − 4x + 12.
x3 − 3x2 − 4x + 12 = 0x2(x− 3)− 4(x− 3) = 0
(x− 3)(x2 − 4) = 0
(x− 3)(x− 2)(x + 2) = 0x = ±2, 3
![Page 17: 3.3 Graphs of Polynomial Functions - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/graphs_of_polynomial_function… · How can we find the roots of a polynomial?](https://reader035.vdocuments.mx/reader035/viewer/2022071212/602443539f43ac2b6a4c4185/html5/thumbnails/17.jpg)
Finding Roots
Example
Find the roots of f (x) = x3 − 3x2 − 4x + 12.
x3 − 3x2 − 4x + 12 = 0x2(x− 3)− 4(x− 3) = 0
(x− 3)(x2 − 4) = 0(x− 3)(x− 2)(x + 2) = 0
x = ±2, 3
![Page 18: 3.3 Graphs of Polynomial Functions - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/graphs_of_polynomial_function… · How can we find the roots of a polynomial?](https://reader035.vdocuments.mx/reader035/viewer/2022071212/602443539f43ac2b6a4c4185/html5/thumbnails/18.jpg)
Finding Roots
Example
Find the roots of f (x) = x3 − 3x2 − 4x + 12.
x3 − 3x2 − 4x + 12 = 0x2(x− 3)− 4(x− 3) = 0
(x− 3)(x2 − 4) = 0(x− 3)(x− 2)(x + 2) = 0
x = ±2, 3
![Page 19: 3.3 Graphs of Polynomial Functions - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/graphs_of_polynomial_function… · How can we find the roots of a polynomial?](https://reader035.vdocuments.mx/reader035/viewer/2022071212/602443539f43ac2b6a4c4185/html5/thumbnails/19.jpg)
An Example
Example
Find the roots of f (x) = x2 − 4x + 4.
x2 − 4x + 4 = 0(x− 2)(x− 2) = 0
x = 2
Anyone remember what a root with multiplicity 2 is called? Adouble root.
![Page 20: 3.3 Graphs of Polynomial Functions - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/graphs_of_polynomial_function… · How can we find the roots of a polynomial?](https://reader035.vdocuments.mx/reader035/viewer/2022071212/602443539f43ac2b6a4c4185/html5/thumbnails/20.jpg)
An Example
Example
Find the roots of f (x) = x2 − 4x + 4.
x2 − 4x + 4 = 0
(x− 2)(x− 2) = 0x = 2
Anyone remember what a root with multiplicity 2 is called? Adouble root.
![Page 21: 3.3 Graphs of Polynomial Functions - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/graphs_of_polynomial_function… · How can we find the roots of a polynomial?](https://reader035.vdocuments.mx/reader035/viewer/2022071212/602443539f43ac2b6a4c4185/html5/thumbnails/21.jpg)
An Example
Example
Find the roots of f (x) = x2 − 4x + 4.
x2 − 4x + 4 = 0(x− 2)(x− 2) = 0
x = 2
Anyone remember what a root with multiplicity 2 is called? Adouble root.
![Page 22: 3.3 Graphs of Polynomial Functions - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/graphs_of_polynomial_function… · How can we find the roots of a polynomial?](https://reader035.vdocuments.mx/reader035/viewer/2022071212/602443539f43ac2b6a4c4185/html5/thumbnails/22.jpg)
An Example
Example
Find the roots of f (x) = x2 − 4x + 4.
x2 − 4x + 4 = 0(x− 2)(x− 2) = 0
x = 2
Anyone remember what a root with multiplicity 2 is called? Adouble root.
![Page 23: 3.3 Graphs of Polynomial Functions - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/graphs_of_polynomial_function… · How can we find the roots of a polynomial?](https://reader035.vdocuments.mx/reader035/viewer/2022071212/602443539f43ac2b6a4c4185/html5/thumbnails/23.jpg)
An Example
Example
Find the roots of f (x) = x2 − 4x + 4.
x2 − 4x + 4 = 0(x− 2)(x− 2) = 0
x = 2
Anyone remember what a root with multiplicity 2 is called?
Adouble root.
![Page 24: 3.3 Graphs of Polynomial Functions - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/graphs_of_polynomial_function… · How can we find the roots of a polynomial?](https://reader035.vdocuments.mx/reader035/viewer/2022071212/602443539f43ac2b6a4c4185/html5/thumbnails/24.jpg)
An Example
Example
Find the roots of f (x) = x2 − 4x + 4.
x2 − 4x + 4 = 0(x− 2)(x− 2) = 0
x = 2
Anyone remember what a root with multiplicity 2 is called? Adouble root.
![Page 25: 3.3 Graphs of Polynomial Functions - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/graphs_of_polynomial_function… · How can we find the roots of a polynomial?](https://reader035.vdocuments.mx/reader035/viewer/2022071212/602443539f43ac2b6a4c4185/html5/thumbnails/25.jpg)
Finding Roots
Example
Find the roots of f (x) = x5 + 3x3 − 4x.
x5 + 3x3 − 4x = 0x(x4 + 3x2 − 4) = 0
x(x2 + 4)(x2 − 1) = 0x(x2 + 4)(x + 1)(x− 1) = 0
x = 0,±1
What about x2 + 4?
![Page 26: 3.3 Graphs of Polynomial Functions - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/graphs_of_polynomial_function… · How can we find the roots of a polynomial?](https://reader035.vdocuments.mx/reader035/viewer/2022071212/602443539f43ac2b6a4c4185/html5/thumbnails/26.jpg)
Finding Roots
Example
Find the roots of f (x) = x5 + 3x3 − 4x.
x5 + 3x3 − 4x = 0
x(x4 + 3x2 − 4) = 0x(x2 + 4)(x2 − 1) = 0
x(x2 + 4)(x + 1)(x− 1) = 0x = 0,±1
What about x2 + 4?
![Page 27: 3.3 Graphs of Polynomial Functions - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/graphs_of_polynomial_function… · How can we find the roots of a polynomial?](https://reader035.vdocuments.mx/reader035/viewer/2022071212/602443539f43ac2b6a4c4185/html5/thumbnails/27.jpg)
Finding Roots
Example
Find the roots of f (x) = x5 + 3x3 − 4x.
x5 + 3x3 − 4x = 0x(x4 + 3x2 − 4) = 0
x(x2 + 4)(x2 − 1) = 0x(x2 + 4)(x + 1)(x− 1) = 0
x = 0,±1
What about x2 + 4?
![Page 28: 3.3 Graphs of Polynomial Functions - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/graphs_of_polynomial_function… · How can we find the roots of a polynomial?](https://reader035.vdocuments.mx/reader035/viewer/2022071212/602443539f43ac2b6a4c4185/html5/thumbnails/28.jpg)
Finding Roots
Example
Find the roots of f (x) = x5 + 3x3 − 4x.
x5 + 3x3 − 4x = 0x(x4 + 3x2 − 4) = 0
x(x2 + 4)(x2 − 1) = 0
x(x2 + 4)(x + 1)(x− 1) = 0x = 0,±1
What about x2 + 4?
![Page 29: 3.3 Graphs of Polynomial Functions - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/graphs_of_polynomial_function… · How can we find the roots of a polynomial?](https://reader035.vdocuments.mx/reader035/viewer/2022071212/602443539f43ac2b6a4c4185/html5/thumbnails/29.jpg)
Finding Roots
Example
Find the roots of f (x) = x5 + 3x3 − 4x.
x5 + 3x3 − 4x = 0x(x4 + 3x2 − 4) = 0
x(x2 + 4)(x2 − 1) = 0x(x2 + 4)(x + 1)(x− 1) = 0
x = 0,±1
What about x2 + 4?
![Page 30: 3.3 Graphs of Polynomial Functions - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/graphs_of_polynomial_function… · How can we find the roots of a polynomial?](https://reader035.vdocuments.mx/reader035/viewer/2022071212/602443539f43ac2b6a4c4185/html5/thumbnails/30.jpg)
Finding Roots
Example
Find the roots of f (x) = x5 + 3x3 − 4x.
x5 + 3x3 − 4x = 0x(x4 + 3x2 − 4) = 0
x(x2 + 4)(x2 − 1) = 0x(x2 + 4)(x + 1)(x− 1) = 0
x = 0,±1
What about x2 + 4?
![Page 31: 3.3 Graphs of Polynomial Functions - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/graphs_of_polynomial_function… · How can we find the roots of a polynomial?](https://reader035.vdocuments.mx/reader035/viewer/2022071212/602443539f43ac2b6a4c4185/html5/thumbnails/31.jpg)
Finding Roots
Example
Find the roots of f (x) = x5 + 3x3 − 4x.
x5 + 3x3 − 4x = 0x(x4 + 3x2 − 4) = 0
x(x2 + 4)(x2 − 1) = 0x(x2 + 4)(x + 1)(x− 1) = 0
x = 0,±1
What about x2 + 4?
![Page 32: 3.3 Graphs of Polynomial Functions - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/graphs_of_polynomial_function… · How can we find the roots of a polynomial?](https://reader035.vdocuments.mx/reader035/viewer/2022071212/602443539f43ac2b6a4c4185/html5/thumbnails/32.jpg)
Roots
Why are roots important?
The roots are the horizontal intercepts (x-intercepts) of thepolynomial. We need them to accurately plot the polynomials.
We also want to be able to find the vertical intercept - how dowe do this?
Example
Find the vertical intercept of f (x) = 3x2 − 3x + 1.
f (0) = 1, so the vertical intercept is (0, 1).
![Page 33: 3.3 Graphs of Polynomial Functions - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/graphs_of_polynomial_function… · How can we find the roots of a polynomial?](https://reader035.vdocuments.mx/reader035/viewer/2022071212/602443539f43ac2b6a4c4185/html5/thumbnails/33.jpg)
Roots
Why are roots important?
The roots are the horizontal intercepts (x-intercepts) of thepolynomial. We need them to accurately plot the polynomials.
We also want to be able to find the vertical intercept - how dowe do this?
Example
Find the vertical intercept of f (x) = 3x2 − 3x + 1.
f (0) = 1, so the vertical intercept is (0, 1).
![Page 34: 3.3 Graphs of Polynomial Functions - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/graphs_of_polynomial_function… · How can we find the roots of a polynomial?](https://reader035.vdocuments.mx/reader035/viewer/2022071212/602443539f43ac2b6a4c4185/html5/thumbnails/34.jpg)
Roots
Why are roots important?
The roots are the horizontal intercepts (x-intercepts) of thepolynomial. We need them to accurately plot the polynomials.
We also want to be able to find the vertical intercept - how dowe do this?
Example
Find the vertical intercept of f (x) = 3x2 − 3x + 1.
f (0) = 1, so the vertical intercept is (0, 1).
![Page 35: 3.3 Graphs of Polynomial Functions - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/graphs_of_polynomial_function… · How can we find the roots of a polynomial?](https://reader035.vdocuments.mx/reader035/viewer/2022071212/602443539f43ac2b6a4c4185/html5/thumbnails/35.jpg)
Roots
Why are roots important?
The roots are the horizontal intercepts (x-intercepts) of thepolynomial. We need them to accurately plot the polynomials.
We also want to be able to find the vertical intercept - how dowe do this?
Example
Find the vertical intercept of f (x) = 3x2 − 3x + 1.
f (0) = 1, so the vertical intercept is (0, 1).
![Page 36: 3.3 Graphs of Polynomial Functions - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/graphs_of_polynomial_function… · How can we find the roots of a polynomial?](https://reader035.vdocuments.mx/reader035/viewer/2022071212/602443539f43ac2b6a4c4185/html5/thumbnails/36.jpg)
Roots
Why are roots important?
The roots are the horizontal intercepts (x-intercepts) of thepolynomial. We need them to accurately plot the polynomials.
We also want to be able to find the vertical intercept - how dowe do this?
Example
Find the vertical intercept of f (x) = 3x2 − 3x + 1.
f (0) = 1, so the vertical intercept is (0, 1).
![Page 37: 3.3 Graphs of Polynomial Functions - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/graphs_of_polynomial_function… · How can we find the roots of a polynomial?](https://reader035.vdocuments.mx/reader035/viewer/2022071212/602443539f43ac2b6a4c4185/html5/thumbnails/37.jpg)
More Roots
What other methods do we know for finding roots?
Quadratic formula (only for quadratics)Completing the square (only for quadratics)Technology
![Page 38: 3.3 Graphs of Polynomial Functions - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/graphs_of_polynomial_function… · How can we find the roots of a polynomial?](https://reader035.vdocuments.mx/reader035/viewer/2022071212/602443539f43ac2b6a4c4185/html5/thumbnails/38.jpg)
More Roots
What other methods do we know for finding roots?
Quadratic formula (only for quadratics)Completing the square (only for quadratics)Technology
![Page 39: 3.3 Graphs of Polynomial Functions - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/graphs_of_polynomial_function… · How can we find the roots of a polynomial?](https://reader035.vdocuments.mx/reader035/viewer/2022071212/602443539f43ac2b6a4c4185/html5/thumbnails/39.jpg)
Pictures and Roots
Single Root
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
4
3
2
1
-4
-3
-2
-1
![Page 40: 3.3 Graphs of Polynomial Functions - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/graphs_of_polynomial_function… · How can we find the roots of a polynomial?](https://reader035.vdocuments.mx/reader035/viewer/2022071212/602443539f43ac2b6a4c4185/html5/thumbnails/40.jpg)
Pictures and Roots
Double Root
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
4
3
2
1
-4
-3
-2
-1
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Pictures and Roots
Triple Root
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
4
3
2
1
-4
-3
-2
-1
![Page 42: 3.3 Graphs of Polynomial Functions - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/graphs_of_polynomial_function… · How can we find the roots of a polynomial?](https://reader035.vdocuments.mx/reader035/viewer/2022071212/602443539f43ac2b6a4c4185/html5/thumbnails/42.jpg)
End Behavior
Our concern here is what happens as x→ ∞ and as x→ −∞.
Example
What is the end behavior of f (x) = x2 − 2x + 1.
What happens to f as x→ ∞? f (x)→ ∞.
What happens to f as x→ −∞? f (x)→ ∞.
![Page 43: 3.3 Graphs of Polynomial Functions - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/graphs_of_polynomial_function… · How can we find the roots of a polynomial?](https://reader035.vdocuments.mx/reader035/viewer/2022071212/602443539f43ac2b6a4c4185/html5/thumbnails/43.jpg)
End Behavior
Our concern here is what happens as x→ ∞ and as x→ −∞.
Example
What is the end behavior of f (x) = x2 − 2x + 1.
What happens to f as x→ ∞? f (x)→ ∞.
What happens to f as x→ −∞? f (x)→ ∞.
![Page 44: 3.3 Graphs of Polynomial Functions - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/graphs_of_polynomial_function… · How can we find the roots of a polynomial?](https://reader035.vdocuments.mx/reader035/viewer/2022071212/602443539f43ac2b6a4c4185/html5/thumbnails/44.jpg)
End Behavior
Our concern here is what happens as x→ ∞ and as x→ −∞.
Example
What is the end behavior of f (x) = x2 − 2x + 1.
What happens to f as x→ ∞?
f (x)→ ∞.
What happens to f as x→ −∞? f (x)→ ∞.
![Page 45: 3.3 Graphs of Polynomial Functions - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/graphs_of_polynomial_function… · How can we find the roots of a polynomial?](https://reader035.vdocuments.mx/reader035/viewer/2022071212/602443539f43ac2b6a4c4185/html5/thumbnails/45.jpg)
End Behavior
Our concern here is what happens as x→ ∞ and as x→ −∞.
Example
What is the end behavior of f (x) = x2 − 2x + 1.
What happens to f as x→ ∞? f (x)→ ∞.
What happens to f as x→ −∞? f (x)→ ∞.
![Page 46: 3.3 Graphs of Polynomial Functions - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/graphs_of_polynomial_function… · How can we find the roots of a polynomial?](https://reader035.vdocuments.mx/reader035/viewer/2022071212/602443539f43ac2b6a4c4185/html5/thumbnails/46.jpg)
End Behavior
Our concern here is what happens as x→ ∞ and as x→ −∞.
Example
What is the end behavior of f (x) = x2 − 2x + 1.
What happens to f as x→ ∞? f (x)→ ∞.
What happens to f as x→ −∞?
f (x)→ ∞.
![Page 47: 3.3 Graphs of Polynomial Functions - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/graphs_of_polynomial_function… · How can we find the roots of a polynomial?](https://reader035.vdocuments.mx/reader035/viewer/2022071212/602443539f43ac2b6a4c4185/html5/thumbnails/47.jpg)
End Behavior
Our concern here is what happens as x→ ∞ and as x→ −∞.
Example
What is the end behavior of f (x) = x2 − 2x + 1.
What happens to f as x→ ∞? f (x)→ ∞.
What happens to f as x→ −∞? f (x)→ ∞.
![Page 48: 3.3 Graphs of Polynomial Functions - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/graphs_of_polynomial_function… · How can we find the roots of a polynomial?](https://reader035.vdocuments.mx/reader035/viewer/2022071212/602443539f43ac2b6a4c4185/html5/thumbnails/48.jpg)
End Behavior
Our concern here is what happens as x→ ∞ and as x→ −∞.
Example
What is the end behavior of f (x) = −x3 + 3x2.
What happens to f as x→ ∞? f (x)→ −∞.
What happens to f as x→ −∞? f (x)→ ∞.
![Page 49: 3.3 Graphs of Polynomial Functions - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/graphs_of_polynomial_function… · How can we find the roots of a polynomial?](https://reader035.vdocuments.mx/reader035/viewer/2022071212/602443539f43ac2b6a4c4185/html5/thumbnails/49.jpg)
End Behavior
Our concern here is what happens as x→ ∞ and as x→ −∞.
Example
What is the end behavior of f (x) = −x3 + 3x2.
What happens to f as x→ ∞? f (x)→ −∞.
What happens to f as x→ −∞? f (x)→ ∞.
![Page 50: 3.3 Graphs of Polynomial Functions - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/graphs_of_polynomial_function… · How can we find the roots of a polynomial?](https://reader035.vdocuments.mx/reader035/viewer/2022071212/602443539f43ac2b6a4c4185/html5/thumbnails/50.jpg)
End Behavior
Our concern here is what happens as x→ ∞ and as x→ −∞.
Example
What is the end behavior of f (x) = −x3 + 3x2.
What happens to f as x→ ∞?
f (x)→ −∞.
What happens to f as x→ −∞? f (x)→ ∞.
![Page 51: 3.3 Graphs of Polynomial Functions - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/graphs_of_polynomial_function… · How can we find the roots of a polynomial?](https://reader035.vdocuments.mx/reader035/viewer/2022071212/602443539f43ac2b6a4c4185/html5/thumbnails/51.jpg)
End Behavior
Our concern here is what happens as x→ ∞ and as x→ −∞.
Example
What is the end behavior of f (x) = −x3 + 3x2.
What happens to f as x→ ∞? f (x)→ −∞.
What happens to f as x→ −∞? f (x)→ ∞.
![Page 52: 3.3 Graphs of Polynomial Functions - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/graphs_of_polynomial_function… · How can we find the roots of a polynomial?](https://reader035.vdocuments.mx/reader035/viewer/2022071212/602443539f43ac2b6a4c4185/html5/thumbnails/52.jpg)
End Behavior
Our concern here is what happens as x→ ∞ and as x→ −∞.
Example
What is the end behavior of f (x) = −x3 + 3x2.
What happens to f as x→ ∞? f (x)→ −∞.
What happens to f as x→ −∞?
f (x)→ ∞.
![Page 53: 3.3 Graphs of Polynomial Functions - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/graphs_of_polynomial_function… · How can we find the roots of a polynomial?](https://reader035.vdocuments.mx/reader035/viewer/2022071212/602443539f43ac2b6a4c4185/html5/thumbnails/53.jpg)
End Behavior
Our concern here is what happens as x→ ∞ and as x→ −∞.
Example
What is the end behavior of f (x) = −x3 + 3x2.
What happens to f as x→ ∞? f (x)→ −∞.
What happens to f as x→ −∞? f (x)→ ∞.
![Page 54: 3.3 Graphs of Polynomial Functions - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/graphs_of_polynomial_function… · How can we find the roots of a polynomial?](https://reader035.vdocuments.mx/reader035/viewer/2022071212/602443539f43ac2b6a4c4185/html5/thumbnails/54.jpg)
Putting it all together
Example
Plot f (x) = x2 − 5x + 6.
Roots? x = 2, 3, so (2, 0) and (3, 0)
y-intercept? (0, 6)
End behavior?
![Page 55: 3.3 Graphs of Polynomial Functions - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/graphs_of_polynomial_function… · How can we find the roots of a polynomial?](https://reader035.vdocuments.mx/reader035/viewer/2022071212/602443539f43ac2b6a4c4185/html5/thumbnails/55.jpg)
Putting it all together
Example
Plot f (x) = x2 − 5x + 6.
Roots?
x = 2, 3, so (2, 0) and (3, 0)
y-intercept? (0, 6)
End behavior?
![Page 56: 3.3 Graphs of Polynomial Functions - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/graphs_of_polynomial_function… · How can we find the roots of a polynomial?](https://reader035.vdocuments.mx/reader035/viewer/2022071212/602443539f43ac2b6a4c4185/html5/thumbnails/56.jpg)
Putting it all together
Example
Plot f (x) = x2 − 5x + 6.
Roots? x = 2, 3, so (2, 0) and (3, 0)
y-intercept? (0, 6)
End behavior?
![Page 57: 3.3 Graphs of Polynomial Functions - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/graphs_of_polynomial_function… · How can we find the roots of a polynomial?](https://reader035.vdocuments.mx/reader035/viewer/2022071212/602443539f43ac2b6a4c4185/html5/thumbnails/57.jpg)
Putting it all together
Example
Plot f (x) = x2 − 5x + 6.
Roots? x = 2, 3, so (2, 0) and (3, 0)
y-intercept?
(0, 6)
End behavior?
![Page 58: 3.3 Graphs of Polynomial Functions - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/graphs_of_polynomial_function… · How can we find the roots of a polynomial?](https://reader035.vdocuments.mx/reader035/viewer/2022071212/602443539f43ac2b6a4c4185/html5/thumbnails/58.jpg)
Putting it all together
Example
Plot f (x) = x2 − 5x + 6.
Roots? x = 2, 3, so (2, 0) and (3, 0)
y-intercept? (0, 6)
End behavior?
![Page 59: 3.3 Graphs of Polynomial Functions - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/graphs_of_polynomial_function… · How can we find the roots of a polynomial?](https://reader035.vdocuments.mx/reader035/viewer/2022071212/602443539f43ac2b6a4c4185/html5/thumbnails/59.jpg)
Putting it all together
Example
Plot f (x) = x2 − 5x + 6.
Roots? x = 2, 3, so (2, 0) and (3, 0)
y-intercept? (0, 6)
End behavior?
![Page 60: 3.3 Graphs of Polynomial Functions - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/graphs_of_polynomial_function… · How can we find the roots of a polynomial?](https://reader035.vdocuments.mx/reader035/viewer/2022071212/602443539f43ac2b6a4c4185/html5/thumbnails/60.jpg)
Putting it all together
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
6
5
4
3
2
1
-4
-3
-2
-1
••
•
•
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Putting it all together
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
6
5
4
3
2
1
-4
-3
-2
-1
••
•
•
![Page 62: 3.3 Graphs of Polynomial Functions - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/graphs_of_polynomial_function… · How can we find the roots of a polynomial?](https://reader035.vdocuments.mx/reader035/viewer/2022071212/602443539f43ac2b6a4c4185/html5/thumbnails/62.jpg)
Putting it all together
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
6
5
4
3
2
1
-4
-3
-2
-1
••
•
•
![Page 63: 3.3 Graphs of Polynomial Functions - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/graphs_of_polynomial_function… · How can we find the roots of a polynomial?](https://reader035.vdocuments.mx/reader035/viewer/2022071212/602443539f43ac2b6a4c4185/html5/thumbnails/63.jpg)
Graphing Polynomials
Example
Plot f (x) = (x− 2)(x− 1)(x + 2).
Roots? (2, 0), (1, 0), (−2, 0).
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
4
3
2
1
-4
-3
-2
-1
•••
![Page 64: 3.3 Graphs of Polynomial Functions - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/graphs_of_polynomial_function… · How can we find the roots of a polynomial?](https://reader035.vdocuments.mx/reader035/viewer/2022071212/602443539f43ac2b6a4c4185/html5/thumbnails/64.jpg)
Graphing Polynomials
Example
Plot f (x) = (x− 2)(x− 1)(x + 2).
Roots?
(2, 0), (1, 0), (−2, 0).
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
4
3
2
1
-4
-3
-2
-1
•••
![Page 65: 3.3 Graphs of Polynomial Functions - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/graphs_of_polynomial_function… · How can we find the roots of a polynomial?](https://reader035.vdocuments.mx/reader035/viewer/2022071212/602443539f43ac2b6a4c4185/html5/thumbnails/65.jpg)
Graphing Polynomials
Example
Plot f (x) = (x− 2)(x− 1)(x + 2).
Roots? (2, 0), (1, 0), (−2, 0).
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
4
3
2
1
-4
-3
-2
-1
•••
![Page 66: 3.3 Graphs of Polynomial Functions - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/graphs_of_polynomial_function… · How can we find the roots of a polynomial?](https://reader035.vdocuments.mx/reader035/viewer/2022071212/602443539f43ac2b6a4c4185/html5/thumbnails/66.jpg)
Graphing Polynomials
Example
Plot f (x) = (x− 2)(x− 1)(x + 2).
y-intercept? f (0) = (0− 2)(0− 1)(0 + 2) = 4, so we have (0, 4)
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
4
3
2
1
-4
-3
-2
-1
•••
•
![Page 67: 3.3 Graphs of Polynomial Functions - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/graphs_of_polynomial_function… · How can we find the roots of a polynomial?](https://reader035.vdocuments.mx/reader035/viewer/2022071212/602443539f43ac2b6a4c4185/html5/thumbnails/67.jpg)
Graphing Polynomials
Example
Plot f (x) = (x− 2)(x− 1)(x + 2).
y-intercept?
f (0) = (0− 2)(0− 1)(0 + 2) = 4, so we have (0, 4)
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
4
3
2
1
-4
-3
-2
-1
•••
•
![Page 68: 3.3 Graphs of Polynomial Functions - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/graphs_of_polynomial_function… · How can we find the roots of a polynomial?](https://reader035.vdocuments.mx/reader035/viewer/2022071212/602443539f43ac2b6a4c4185/html5/thumbnails/68.jpg)
Graphing Polynomials
Example
Plot f (x) = (x− 2)(x− 1)(x + 2).
y-intercept? f (0) = (0− 2)(0− 1)(0 + 2) = 4, so we have (0, 4)
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
4
3
2
1
-4
-3
-2
-1
•••
•
![Page 69: 3.3 Graphs of Polynomial Functions - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/graphs_of_polynomial_function… · How can we find the roots of a polynomial?](https://reader035.vdocuments.mx/reader035/viewer/2022071212/602443539f43ac2b6a4c4185/html5/thumbnails/69.jpg)
Graphing Polynomials
Example
Plot f (x) = (x− 2)(x− 1)(x + 2).
y-intercept? f (0) = (0− 2)(0− 1)(0 + 2) = 4, so we have (0, 4)
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
4
3
2
1
-4
-3
-2
-1
•••
•
![Page 70: 3.3 Graphs of Polynomial Functions - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/graphs_of_polynomial_function… · How can we find the roots of a polynomial?](https://reader035.vdocuments.mx/reader035/viewer/2022071212/602443539f43ac2b6a4c4185/html5/thumbnails/70.jpg)
Graphing Polynomials
Example
Plot f (x) = (x− 2)(x− 1)(x + 2).
End behavior?
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
4
3
2
1
-4
-3
-2
-1
•••
•
![Page 71: 3.3 Graphs of Polynomial Functions - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/graphs_of_polynomial_function… · How can we find the roots of a polynomial?](https://reader035.vdocuments.mx/reader035/viewer/2022071212/602443539f43ac2b6a4c4185/html5/thumbnails/71.jpg)
Graphing Polynomials
Example
Plot f (x) = (x− 2)(x− 1)(x + 2).
End behavior?
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
4
3
2
1
-4
-3
-2
-1
•••
•
![Page 72: 3.3 Graphs of Polynomial Functions - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/graphs_of_polynomial_function… · How can we find the roots of a polynomial?](https://reader035.vdocuments.mx/reader035/viewer/2022071212/602443539f43ac2b6a4c4185/html5/thumbnails/72.jpg)
Graphing Polynomials
Example
Plot f (x) = (x− 2)(x− 1)(x + 2).
End behavior?
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
4
3
2
1
-4
-3
-2
-1
•••
•
![Page 73: 3.3 Graphs of Polynomial Functions - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/graphs_of_polynomial_function… · How can we find the roots of a polynomial?](https://reader035.vdocuments.mx/reader035/viewer/2022071212/602443539f43ac2b6a4c4185/html5/thumbnails/73.jpg)
Graphing Polynomials
Example
Graph g(x) = (x− 1)(x + 2)2.
Roots? (1, 0) and (−2, 0) with multiplicity 2.
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
4
3
2
1
-4
-3
-2
-1
• •
![Page 74: 3.3 Graphs of Polynomial Functions - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/graphs_of_polynomial_function… · How can we find the roots of a polynomial?](https://reader035.vdocuments.mx/reader035/viewer/2022071212/602443539f43ac2b6a4c4185/html5/thumbnails/74.jpg)
Graphing Polynomials
Example
Graph g(x) = (x− 1)(x + 2)2.
Roots?
(1, 0) and (−2, 0) with multiplicity 2.
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
4
3
2
1
-4
-3
-2
-1
• •
![Page 75: 3.3 Graphs of Polynomial Functions - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/graphs_of_polynomial_function… · How can we find the roots of a polynomial?](https://reader035.vdocuments.mx/reader035/viewer/2022071212/602443539f43ac2b6a4c4185/html5/thumbnails/75.jpg)
Graphing Polynomials
Example
Graph g(x) = (x− 1)(x + 2)2.
Roots? (1, 0) and (−2, 0)
with multiplicity 2.
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
4
3
2
1
-4
-3
-2
-1
• •
![Page 76: 3.3 Graphs of Polynomial Functions - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/graphs_of_polynomial_function… · How can we find the roots of a polynomial?](https://reader035.vdocuments.mx/reader035/viewer/2022071212/602443539f43ac2b6a4c4185/html5/thumbnails/76.jpg)
Graphing Polynomials
Example
Graph g(x) = (x− 1)(x + 2)2.
Roots? (1, 0) and (−2, 0) with multiplicity 2.
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
4
3
2
1
-4
-3
-2
-1
• •
![Page 77: 3.3 Graphs of Polynomial Functions - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/graphs_of_polynomial_function… · How can we find the roots of a polynomial?](https://reader035.vdocuments.mx/reader035/viewer/2022071212/602443539f43ac2b6a4c4185/html5/thumbnails/77.jpg)
Graphing Polynomials
Example
Graph g(x) = (x− 1)(x + 2)2.
Roots? (1, 0) and (−2, 0) with multiplicity 2.
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
4
3
2
1
-4
-3
-2
-1
• •
![Page 78: 3.3 Graphs of Polynomial Functions - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/graphs_of_polynomial_function… · How can we find the roots of a polynomial?](https://reader035.vdocuments.mx/reader035/viewer/2022071212/602443539f43ac2b6a4c4185/html5/thumbnails/78.jpg)
Graphing Polynomials
Example
Graph g(x) = (x− 1)(x + 2)2.
y-intercept? g(0) = (0− 1)(0 + 2)2 = −4 so (0,−4).
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
4
3
2
1
-4
-3
-2
-1
• •
•
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Graphing Polynomials
Example
Graph g(x) = (x− 1)(x + 2)2.
y-intercept?
g(0) = (0− 1)(0 + 2)2 = −4 so (0,−4).
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
4
3
2
1
-4
-3
-2
-1
• •
•
![Page 80: 3.3 Graphs of Polynomial Functions - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/graphs_of_polynomial_function… · How can we find the roots of a polynomial?](https://reader035.vdocuments.mx/reader035/viewer/2022071212/602443539f43ac2b6a4c4185/html5/thumbnails/80.jpg)
Graphing Polynomials
Example
Graph g(x) = (x− 1)(x + 2)2.
y-intercept? g(0) = (0− 1)(0 + 2)2 = −4 so (0,−4).
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
4
3
2
1
-4
-3
-2
-1
• •
•
![Page 81: 3.3 Graphs of Polynomial Functions - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/graphs_of_polynomial_function… · How can we find the roots of a polynomial?](https://reader035.vdocuments.mx/reader035/viewer/2022071212/602443539f43ac2b6a4c4185/html5/thumbnails/81.jpg)
Graphing Polynomials
Example
Graph g(x) = (x− 1)(x + 2)2.
y-intercept? g(0) = (0− 1)(0 + 2)2 = −4 so (0,−4).
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
4
3
2
1
-4
-3
-2
-1
• •
•
![Page 82: 3.3 Graphs of Polynomial Functions - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/graphs_of_polynomial_function… · How can we find the roots of a polynomial?](https://reader035.vdocuments.mx/reader035/viewer/2022071212/602443539f43ac2b6a4c4185/html5/thumbnails/82.jpg)
Graphing Polynomials
Example
Graph g(x) = (x− 1)(x + 2)2.
End behavior?
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
4
3
2
1
-4
-3
-2
-1
• •
•
![Page 83: 3.3 Graphs of Polynomial Functions - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/graphs_of_polynomial_function… · How can we find the roots of a polynomial?](https://reader035.vdocuments.mx/reader035/viewer/2022071212/602443539f43ac2b6a4c4185/html5/thumbnails/83.jpg)
Graphing Polynomials
Example
Graph g(x) = (x− 1)(x + 2)2.
End behavior?
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
4
3
2
1
-4
-3
-2
-1
• •
•
![Page 84: 3.3 Graphs of Polynomial Functions - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/graphs_of_polynomial_function… · How can we find the roots of a polynomial?](https://reader035.vdocuments.mx/reader035/viewer/2022071212/602443539f43ac2b6a4c4185/html5/thumbnails/84.jpg)
Graphing Polynomials
Example
Graph g(x) = (x− 1)(x + 2)2.
End behavior?
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
4
3
2
1
-4
-3
-2
-1
• •
•
![Page 85: 3.3 Graphs of Polynomial Functions - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/graphs_of_polynomial_function… · How can we find the roots of a polynomial?](https://reader035.vdocuments.mx/reader035/viewer/2022071212/602443539f43ac2b6a4c4185/html5/thumbnails/85.jpg)
Finding the Polynomial
Example
Find the equation of a polynomial with roots at x = ±1, 3 andwith y-intercept at (0, 2).
We know we have
f (x) = a(x− 1)(x + 1)(x− 3)
but we need to find the parameter a. How can we do this?
a(0− 1)(0 + 1)(0− 3) = 23a = 2
a =23
So, our equation is
f (x) =23(x− 1)(x + 1)(x− 3)
![Page 86: 3.3 Graphs of Polynomial Functions - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/graphs_of_polynomial_function… · How can we find the roots of a polynomial?](https://reader035.vdocuments.mx/reader035/viewer/2022071212/602443539f43ac2b6a4c4185/html5/thumbnails/86.jpg)
Finding the Polynomial
Example
Find the equation of a polynomial with roots at x = ±1, 3 andwith y-intercept at (0, 2).
We know we have
f (x) = a(x− 1)(x + 1)(x− 3)
but we need to find the parameter a. How can we do this?
a(0− 1)(0 + 1)(0− 3) = 23a = 2
a =23
So, our equation is
f (x) =23(x− 1)(x + 1)(x− 3)
![Page 87: 3.3 Graphs of Polynomial Functions - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/graphs_of_polynomial_function… · How can we find the roots of a polynomial?](https://reader035.vdocuments.mx/reader035/viewer/2022071212/602443539f43ac2b6a4c4185/html5/thumbnails/87.jpg)
Finding the Polynomial
Example
Find the equation of a polynomial with roots at x = ±1, 3 andwith y-intercept at (0, 2).
We know we have
f (x) = a(x− 1)(x + 1)(x− 3)
but we need to find the parameter a. How can we do this?
a(0− 1)(0 + 1)(0− 3) = 2
3a = 2
a =23
So, our equation is
f (x) =23(x− 1)(x + 1)(x− 3)
![Page 88: 3.3 Graphs of Polynomial Functions - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/graphs_of_polynomial_function… · How can we find the roots of a polynomial?](https://reader035.vdocuments.mx/reader035/viewer/2022071212/602443539f43ac2b6a4c4185/html5/thumbnails/88.jpg)
Finding the Polynomial
Example
Find the equation of a polynomial with roots at x = ±1, 3 andwith y-intercept at (0, 2).
We know we have
f (x) = a(x− 1)(x + 1)(x− 3)
but we need to find the parameter a. How can we do this?
a(0− 1)(0 + 1)(0− 3) = 23a = 2
a =23
So, our equation is
f (x) =23(x− 1)(x + 1)(x− 3)
![Page 89: 3.3 Graphs of Polynomial Functions - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/graphs_of_polynomial_function… · How can we find the roots of a polynomial?](https://reader035.vdocuments.mx/reader035/viewer/2022071212/602443539f43ac2b6a4c4185/html5/thumbnails/89.jpg)
Finding the Polynomial
Example
Find the equation of a polynomial with roots at x = ±1, 3 andwith y-intercept at (0, 2).
We know we have
f (x) = a(x− 1)(x + 1)(x− 3)
but we need to find the parameter a. How can we do this?
a(0− 1)(0 + 1)(0− 3) = 23a = 2
a =23
So, our equation is
f (x) =23(x− 1)(x + 1)(x− 3)
![Page 90: 3.3 Graphs of Polynomial Functions - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/graphs_of_polynomial_function… · How can we find the roots of a polynomial?](https://reader035.vdocuments.mx/reader035/viewer/2022071212/602443539f43ac2b6a4c4185/html5/thumbnails/90.jpg)
Finding the Polynomial
Example
Find the equation of a polynomial with roots at x = ±1, 3 andwith y-intercept at (0, 2).
We know we have
f (x) = a(x− 1)(x + 1)(x− 3)
but we need to find the parameter a. How can we do this?
a(0− 1)(0 + 1)(0− 3) = 23a = 2
a =23
So, our equation is
f (x) =23(x− 1)(x + 1)(x− 3)
![Page 91: 3.3 Graphs of Polynomial Functions - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/graphs_of_polynomial_function… · How can we find the roots of a polynomial?](https://reader035.vdocuments.mx/reader035/viewer/2022071212/602443539f43ac2b6a4c4185/html5/thumbnails/91.jpg)
Finding the Polynomial
We can do the same thing from a picture ...
Example
Find an equation for the the polynomial given below.
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
4
3
2
1
-4
-3
-2
-1
• • •
•
![Page 92: 3.3 Graphs of Polynomial Functions - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/graphs_of_polynomial_function… · How can we find the roots of a polynomial?](https://reader035.vdocuments.mx/reader035/viewer/2022071212/602443539f43ac2b6a4c4185/html5/thumbnails/92.jpg)
Finding the Polynomial
Roots?
(3, 0), (1, 0), (−3, 0) with multiplicity 2.
So we have ...
f (x) = a(x− 3)(x− 1)(x + 3)2
Now ...
a(0− 3)(0− 1)(0 + 3)2 = 327a = 3
a =19
So,
f (x) =19(x− 3)(x− 1)(x + 3)2
![Page 93: 3.3 Graphs of Polynomial Functions - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/graphs_of_polynomial_function… · How can we find the roots of a polynomial?](https://reader035.vdocuments.mx/reader035/viewer/2022071212/602443539f43ac2b6a4c4185/html5/thumbnails/93.jpg)
Finding the Polynomial
Roots? (3, 0), (1, 0), (−3, 0) with multiplicity 2.
So we have ...
f (x) = a(x− 3)(x− 1)(x + 3)2
Now ...
a(0− 3)(0− 1)(0 + 3)2 = 327a = 3
a =19
So,
f (x) =19(x− 3)(x− 1)(x + 3)2
![Page 94: 3.3 Graphs of Polynomial Functions - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/graphs_of_polynomial_function… · How can we find the roots of a polynomial?](https://reader035.vdocuments.mx/reader035/viewer/2022071212/602443539f43ac2b6a4c4185/html5/thumbnails/94.jpg)
Finding the Polynomial
Roots? (3, 0), (1, 0), (−3, 0) with multiplicity 2.
So we have ...
f (x) = a(x− 3)(x− 1)(x + 3)2
Now ...
a(0− 3)(0− 1)(0 + 3)2 = 327a = 3
a =19
So,
f (x) =19(x− 3)(x− 1)(x + 3)2
![Page 95: 3.3 Graphs of Polynomial Functions - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/graphs_of_polynomial_function… · How can we find the roots of a polynomial?](https://reader035.vdocuments.mx/reader035/viewer/2022071212/602443539f43ac2b6a4c4185/html5/thumbnails/95.jpg)
Finding the Polynomial
Roots? (3, 0), (1, 0), (−3, 0) with multiplicity 2.
So we have ...
f (x) = a(x− 3)(x− 1)(x + 3)2
Now ...
a(0− 3)(0− 1)(0 + 3)2 = 327a = 3
a =19
So,
f (x) =19(x− 3)(x− 1)(x + 3)2
![Page 96: 3.3 Graphs of Polynomial Functions - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/graphs_of_polynomial_function… · How can we find the roots of a polynomial?](https://reader035.vdocuments.mx/reader035/viewer/2022071212/602443539f43ac2b6a4c4185/html5/thumbnails/96.jpg)
Finding the Polynomial
Roots? (3, 0), (1, 0), (−3, 0) with multiplicity 2.
So we have ...
f (x) = a(x− 3)(x− 1)(x + 3)2
Now ...
a(0− 3)(0− 1)(0 + 3)2 = 327a = 3
a =19
So,
f (x) =19(x− 3)(x− 1)(x + 3)2
![Page 97: 3.3 Graphs of Polynomial Functions - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/graphs_of_polynomial_function… · How can we find the roots of a polynomial?](https://reader035.vdocuments.mx/reader035/viewer/2022071212/602443539f43ac2b6a4c4185/html5/thumbnails/97.jpg)
Finding the Polynomial
Roots? (3, 0), (1, 0), (−3, 0) with multiplicity 2.
So we have ...
f (x) = a(x− 3)(x− 1)(x + 3)2
Now ...
a(0− 3)(0− 1)(0 + 3)2 = 3
27a = 3
a =19
So,
f (x) =19(x− 3)(x− 1)(x + 3)2
![Page 98: 3.3 Graphs of Polynomial Functions - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/graphs_of_polynomial_function… · How can we find the roots of a polynomial?](https://reader035.vdocuments.mx/reader035/viewer/2022071212/602443539f43ac2b6a4c4185/html5/thumbnails/98.jpg)
Finding the Polynomial
Roots? (3, 0), (1, 0), (−3, 0) with multiplicity 2.
So we have ...
f (x) = a(x− 3)(x− 1)(x + 3)2
Now ...
a(0− 3)(0− 1)(0 + 3)2 = 327a = 3
a =19
So,
f (x) =19(x− 3)(x− 1)(x + 3)2
![Page 99: 3.3 Graphs of Polynomial Functions - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/graphs_of_polynomial_function… · How can we find the roots of a polynomial?](https://reader035.vdocuments.mx/reader035/viewer/2022071212/602443539f43ac2b6a4c4185/html5/thumbnails/99.jpg)
Finding the Polynomial
Roots? (3, 0), (1, 0), (−3, 0) with multiplicity 2.
So we have ...
f (x) = a(x− 3)(x− 1)(x + 3)2
Now ...
a(0− 3)(0− 1)(0 + 3)2 = 327a = 3
a =19
So,
f (x) =19(x− 3)(x− 1)(x + 3)2
![Page 100: 3.3 Graphs of Polynomial Functions - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/graphs_of_polynomial_function… · How can we find the roots of a polynomial?](https://reader035.vdocuments.mx/reader035/viewer/2022071212/602443539f43ac2b6a4c4185/html5/thumbnails/100.jpg)
Finding the Polynomial
Roots? (3, 0), (1, 0), (−3, 0) with multiplicity 2.
So we have ...
f (x) = a(x− 3)(x− 1)(x + 3)2
Now ...
a(0− 3)(0− 1)(0 + 3)2 = 327a = 3
a =19
So,
f (x) =19(x− 3)(x− 1)(x + 3)2
![Page 101: 3.3 Graphs of Polynomial Functions - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/graphs_of_polynomial_function… · How can we find the roots of a polynomial?](https://reader035.vdocuments.mx/reader035/viewer/2022071212/602443539f43ac2b6a4c4185/html5/thumbnails/101.jpg)
Solving Inequalities
One more skill we need from here ...
Example
Solve (x− 4)(x− 1)(x + 3) > 0
What are the roots? x = 4, 1,−3
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6• • • •
![Page 102: 3.3 Graphs of Polynomial Functions - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/graphs_of_polynomial_function… · How can we find the roots of a polynomial?](https://reader035.vdocuments.mx/reader035/viewer/2022071212/602443539f43ac2b6a4c4185/html5/thumbnails/102.jpg)
Solving Inequalities
One more skill we need from here ...
Example
Solve (x− 4)(x− 1)(x + 3) > 0
What are the roots? x = 4, 1,−3
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6• • • •
![Page 103: 3.3 Graphs of Polynomial Functions - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/graphs_of_polynomial_function… · How can we find the roots of a polynomial?](https://reader035.vdocuments.mx/reader035/viewer/2022071212/602443539f43ac2b6a4c4185/html5/thumbnails/103.jpg)
Solving Inequalities
One more skill we need from here ...
Example
Solve (x− 4)(x− 1)(x + 3) > 0
What are the roots?
x = 4, 1,−3
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6• • • •
![Page 104: 3.3 Graphs of Polynomial Functions - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/graphs_of_polynomial_function… · How can we find the roots of a polynomial?](https://reader035.vdocuments.mx/reader035/viewer/2022071212/602443539f43ac2b6a4c4185/html5/thumbnails/104.jpg)
Solving Inequalities
One more skill we need from here ...
Example
Solve (x− 4)(x− 1)(x + 3) > 0
What are the roots? x = 4, 1,−3
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6• • • •
![Page 105: 3.3 Graphs of Polynomial Functions - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/graphs_of_polynomial_function… · How can we find the roots of a polynomial?](https://reader035.vdocuments.mx/reader035/viewer/2022071212/602443539f43ac2b6a4c4185/html5/thumbnails/105.jpg)
Solving Inequalities
One more skill we need from here ...
Example
Solve (x− 4)(x− 1)(x + 3) > 0
What are the roots? x = 4, 1,−3
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
• • • •
![Page 106: 3.3 Graphs of Polynomial Functions - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/graphs_of_polynomial_function… · How can we find the roots of a polynomial?](https://reader035.vdocuments.mx/reader035/viewer/2022071212/602443539f43ac2b6a4c4185/html5/thumbnails/106.jpg)
Solving Inequalities
One more skill we need from here ...
Example
Solve (x− 4)(x− 1)(x + 3) > 0
What are the roots? x = 4, 1,−3
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6• • • •
![Page 107: 3.3 Graphs of Polynomial Functions - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/graphs_of_polynomial_function… · How can we find the roots of a polynomial?](https://reader035.vdocuments.mx/reader035/viewer/2022071212/602443539f43ac2b6a4c4185/html5/thumbnails/107.jpg)
Solving Inequalities
f (−4) = (−4− 4)(−4− 1)(−4 + 3)
f (−4) = (−)(−)(−) < 0f (−1) = (−1− 4)(−1− 1)(−1 + 3)
f (−1) = (−)(−)(+) > 0f (2) = (2− 4)(2− 1)(2 + 3)
f (2) = (−)(+)(+) < 0f (5) = (5− 4)(5− 1)(5 + 3)
f (5) = (+)(+)(+) > 0
![Page 108: 3.3 Graphs of Polynomial Functions - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/graphs_of_polynomial_function… · How can we find the roots of a polynomial?](https://reader035.vdocuments.mx/reader035/viewer/2022071212/602443539f43ac2b6a4c4185/html5/thumbnails/108.jpg)
Solving Inequalities
f (−4) = (−4− 4)(−4− 1)(−4 + 3)f (−4) = (−)(−)(−)
< 0f (−1) = (−1− 4)(−1− 1)(−1 + 3)
f (−1) = (−)(−)(+) > 0f (2) = (2− 4)(2− 1)(2 + 3)
f (2) = (−)(+)(+) < 0f (5) = (5− 4)(5− 1)(5 + 3)
f (5) = (+)(+)(+) > 0
![Page 109: 3.3 Graphs of Polynomial Functions - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/graphs_of_polynomial_function… · How can we find the roots of a polynomial?](https://reader035.vdocuments.mx/reader035/viewer/2022071212/602443539f43ac2b6a4c4185/html5/thumbnails/109.jpg)
Solving Inequalities
f (−4) = (−4− 4)(−4− 1)(−4 + 3)f (−4) = (−)(−)(−) < 0
f (−1) = (−1− 4)(−1− 1)(−1 + 3)f (−1) = (−)(−)(+) > 0
f (2) = (2− 4)(2− 1)(2 + 3)f (2) = (−)(+)(+) < 0
f (5) = (5− 4)(5− 1)(5 + 3)f (5) = (+)(+)(+) > 0
![Page 110: 3.3 Graphs of Polynomial Functions - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/graphs_of_polynomial_function… · How can we find the roots of a polynomial?](https://reader035.vdocuments.mx/reader035/viewer/2022071212/602443539f43ac2b6a4c4185/html5/thumbnails/110.jpg)
Solving Inequalities
f (−4) = (−4− 4)(−4− 1)(−4 + 3)f (−4) = (−)(−)(−) < 0
f (−1) = (−1− 4)(−1− 1)(−1 + 3)
f (−1) = (−)(−)(+) > 0f (2) = (2− 4)(2− 1)(2 + 3)
f (2) = (−)(+)(+) < 0f (5) = (5− 4)(5− 1)(5 + 3)
f (5) = (+)(+)(+) > 0
![Page 111: 3.3 Graphs of Polynomial Functions - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/graphs_of_polynomial_function… · How can we find the roots of a polynomial?](https://reader035.vdocuments.mx/reader035/viewer/2022071212/602443539f43ac2b6a4c4185/html5/thumbnails/111.jpg)
Solving Inequalities
f (−4) = (−4− 4)(−4− 1)(−4 + 3)f (−4) = (−)(−)(−) < 0
f (−1) = (−1− 4)(−1− 1)(−1 + 3)f (−1) = (−)(−)(+)
> 0f (2) = (2− 4)(2− 1)(2 + 3)
f (2) = (−)(+)(+) < 0f (5) = (5− 4)(5− 1)(5 + 3)
f (5) = (+)(+)(+) > 0
![Page 112: 3.3 Graphs of Polynomial Functions - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/graphs_of_polynomial_function… · How can we find the roots of a polynomial?](https://reader035.vdocuments.mx/reader035/viewer/2022071212/602443539f43ac2b6a4c4185/html5/thumbnails/112.jpg)
Solving Inequalities
f (−4) = (−4− 4)(−4− 1)(−4 + 3)f (−4) = (−)(−)(−) < 0
f (−1) = (−1− 4)(−1− 1)(−1 + 3)f (−1) = (−)(−)(+) > 0
f (2) = (2− 4)(2− 1)(2 + 3)f (2) = (−)(+)(+) < 0
f (5) = (5− 4)(5− 1)(5 + 3)f (5) = (+)(+)(+) > 0
![Page 113: 3.3 Graphs of Polynomial Functions - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/graphs_of_polynomial_function… · How can we find the roots of a polynomial?](https://reader035.vdocuments.mx/reader035/viewer/2022071212/602443539f43ac2b6a4c4185/html5/thumbnails/113.jpg)
Solving Inequalities
f (−4) = (−4− 4)(−4− 1)(−4 + 3)f (−4) = (−)(−)(−) < 0
f (−1) = (−1− 4)(−1− 1)(−1 + 3)f (−1) = (−)(−)(+) > 0
f (2) = (2− 4)(2− 1)(2 + 3)
f (2) = (−)(+)(+) < 0f (5) = (5− 4)(5− 1)(5 + 3)
f (5) = (+)(+)(+) > 0
![Page 114: 3.3 Graphs of Polynomial Functions - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/graphs_of_polynomial_function… · How can we find the roots of a polynomial?](https://reader035.vdocuments.mx/reader035/viewer/2022071212/602443539f43ac2b6a4c4185/html5/thumbnails/114.jpg)
Solving Inequalities
f (−4) = (−4− 4)(−4− 1)(−4 + 3)f (−4) = (−)(−)(−) < 0
f (−1) = (−1− 4)(−1− 1)(−1 + 3)f (−1) = (−)(−)(+) > 0
f (2) = (2− 4)(2− 1)(2 + 3)f (2) = (−)(+)(+)
< 0f (5) = (5− 4)(5− 1)(5 + 3)
f (5) = (+)(+)(+) > 0
![Page 115: 3.3 Graphs of Polynomial Functions - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/graphs_of_polynomial_function… · How can we find the roots of a polynomial?](https://reader035.vdocuments.mx/reader035/viewer/2022071212/602443539f43ac2b6a4c4185/html5/thumbnails/115.jpg)
Solving Inequalities
f (−4) = (−4− 4)(−4− 1)(−4 + 3)f (−4) = (−)(−)(−) < 0
f (−1) = (−1− 4)(−1− 1)(−1 + 3)f (−1) = (−)(−)(+) > 0
f (2) = (2− 4)(2− 1)(2 + 3)f (2) = (−)(+)(+) < 0
f (5) = (5− 4)(5− 1)(5 + 3)f (5) = (+)(+)(+) > 0
![Page 116: 3.3 Graphs of Polynomial Functions - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/graphs_of_polynomial_function… · How can we find the roots of a polynomial?](https://reader035.vdocuments.mx/reader035/viewer/2022071212/602443539f43ac2b6a4c4185/html5/thumbnails/116.jpg)
Solving Inequalities
f (−4) = (−4− 4)(−4− 1)(−4 + 3)f (−4) = (−)(−)(−) < 0
f (−1) = (−1− 4)(−1− 1)(−1 + 3)f (−1) = (−)(−)(+) > 0
f (2) = (2− 4)(2− 1)(2 + 3)f (2) = (−)(+)(+) < 0
f (5) = (5− 4)(5− 1)(5 + 3)
f (5) = (+)(+)(+) > 0
![Page 117: 3.3 Graphs of Polynomial Functions - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/graphs_of_polynomial_function… · How can we find the roots of a polynomial?](https://reader035.vdocuments.mx/reader035/viewer/2022071212/602443539f43ac2b6a4c4185/html5/thumbnails/117.jpg)
Solving Inequalities
f (−4) = (−4− 4)(−4− 1)(−4 + 3)f (−4) = (−)(−)(−) < 0
f (−1) = (−1− 4)(−1− 1)(−1 + 3)f (−1) = (−)(−)(+) > 0
f (2) = (2− 4)(2− 1)(2 + 3)f (2) = (−)(+)(+) < 0
f (5) = (5− 4)(5− 1)(5 + 3)f (5) = (+)(+)(+)
> 0
![Page 118: 3.3 Graphs of Polynomial Functions - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/graphs_of_polynomial_function… · How can we find the roots of a polynomial?](https://reader035.vdocuments.mx/reader035/viewer/2022071212/602443539f43ac2b6a4c4185/html5/thumbnails/118.jpg)
Solving Inequalities
f (−4) = (−4− 4)(−4− 1)(−4 + 3)f (−4) = (−)(−)(−) < 0
f (−1) = (−1− 4)(−1− 1)(−1 + 3)f (−1) = (−)(−)(+) > 0
f (2) = (2− 4)(2− 1)(2 + 3)f (2) = (−)(+)(+) < 0
f (5) = (5− 4)(5− 1)(5 + 3)f (5) = (+)(+)(+) > 0
![Page 119: 3.3 Graphs of Polynomial Functions - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/graphs_of_polynomial_function… · How can we find the roots of a polynomial?](https://reader035.vdocuments.mx/reader035/viewer/2022071212/602443539f43ac2b6a4c4185/html5/thumbnails/119.jpg)
Solving Inequalities
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6• • • •
f (x)
f (x)
f (x)
f (x)
So, (x− 4)(x− 1)(x + 3) > 0 on (−3, 1) ∪ (4, ∞)
![Page 120: 3.3 Graphs of Polynomial Functions - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/graphs_of_polynomial_function… · How can we find the roots of a polynomial?](https://reader035.vdocuments.mx/reader035/viewer/2022071212/602443539f43ac2b6a4c4185/html5/thumbnails/120.jpg)
Solving Inequalities
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6• • • •
f (x)
f (x)
f (x)
f (x)
So, (x− 4)(x− 1)(x + 3) > 0 on (−3, 1) ∪ (4, ∞)
![Page 121: 3.3 Graphs of Polynomial Functions - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/graphs_of_polynomial_function… · How can we find the roots of a polynomial?](https://reader035.vdocuments.mx/reader035/viewer/2022071212/602443539f43ac2b6a4c4185/html5/thumbnails/121.jpg)
Last Example
Example
Solve (x + 2)(x− 1)2 > 0
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6• • •
f (x)
f (x) f (x)
![Page 122: 3.3 Graphs of Polynomial Functions - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/graphs_of_polynomial_function… · How can we find the roots of a polynomial?](https://reader035.vdocuments.mx/reader035/viewer/2022071212/602443539f43ac2b6a4c4185/html5/thumbnails/122.jpg)
Last Example
Example
Solve (x + 2)(x− 1)2 > 0
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
• • •
f (x)
f (x) f (x)
![Page 123: 3.3 Graphs of Polynomial Functions - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/graphs_of_polynomial_function… · How can we find the roots of a polynomial?](https://reader035.vdocuments.mx/reader035/viewer/2022071212/602443539f43ac2b6a4c4185/html5/thumbnails/123.jpg)
Last Example
Example
Solve (x + 2)(x− 1)2 > 0
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6• • •
f (x)
f (x) f (x)
![Page 124: 3.3 Graphs of Polynomial Functions - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/graphs_of_polynomial_function… · How can we find the roots of a polynomial?](https://reader035.vdocuments.mx/reader035/viewer/2022071212/602443539f43ac2b6a4c4185/html5/thumbnails/124.jpg)
Last Example
Example
Solve (x + 2)(x− 1)2 > 0
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6• • •
f (x)
f (x) f (x)
![Page 125: 3.3 Graphs of Polynomial Functions - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/graphs_of_polynomial_function… · How can we find the roots of a polynomial?](https://reader035.vdocuments.mx/reader035/viewer/2022071212/602443539f43ac2b6a4c4185/html5/thumbnails/125.jpg)
Last Example
Example
Solve (x + 2)(x− 1)2 > 0
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6• • •
f (x)
f (x)
f (x)
![Page 126: 3.3 Graphs of Polynomial Functions - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/graphs_of_polynomial_function… · How can we find the roots of a polynomial?](https://reader035.vdocuments.mx/reader035/viewer/2022071212/602443539f43ac2b6a4c4185/html5/thumbnails/126.jpg)
Last Example
Example
Solve (x + 2)(x− 1)2 > 0
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6• • •
f (x)
f (x) f (x)
![Page 127: 3.3 Graphs of Polynomial Functions - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/graphs_of_polynomial_function… · How can we find the roots of a polynomial?](https://reader035.vdocuments.mx/reader035/viewer/2022071212/602443539f43ac2b6a4c4185/html5/thumbnails/127.jpg)
Last Example
Example
Solve (x + 2)(x− 1)2 > 0
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6• • •
f (x)
f (x) f (x)
(x + 2)(x− 1)2 > 0 on the interval
(−2, 1) ∪ (1, ∞)
![Page 128: 3.3 Graphs of Polynomial Functions - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/graphs_of_polynomial_function… · How can we find the roots of a polynomial?](https://reader035.vdocuments.mx/reader035/viewer/2022071212/602443539f43ac2b6a4c4185/html5/thumbnails/128.jpg)
Last Example
Example
Solve (x + 2)(x− 1)2 > 0
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6• • •
f (x)
f (x) f (x)
(x + 2)(x− 1)2 > 0 on the interval (−2, 1) ∪ (1, ∞)