solving polynomial equations ppt 5.3.2. factor polynomial expressions in the previous lesson, you...
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![Page 1: Solving Polynomial Equations PPT 5.3.2. Factor Polynomial Expressions In the previous lesson, you factored various polynomial expressions. Such as: x](https://reader036.vdocuments.net/reader036/viewer/2022082516/56649d835503460f94a69fa5/html5/thumbnails/1.jpg)
Solving Polynomial Solving Polynomial EquationsEquations
PPT 5.3.2
![Page 2: Solving Polynomial Equations PPT 5.3.2. Factor Polynomial Expressions In the previous lesson, you factored various polynomial expressions. Such as: x](https://reader036.vdocuments.net/reader036/viewer/2022082516/56649d835503460f94a69fa5/html5/thumbnails/2.jpg)
Factor Polynomial Expressions
In the previous lesson, you factored various polynomial expressions.
Such as:x3 – 2x2 =x4 – x3 – 3x2 + 3x =
= =
Common Factor
x2(x – 2)
x[x2(x – 1) – 3(x – 1)]
x(x2 – 3)(x – 1)
x(x3 – x2 – 3x + 3)
Grouping – common factor the first two terms and then the last two terms.
Common Factor
Refer to 5.2.2 in Lesson 2 to review which strategy is required for each question.
Refer to 5.2.2 in Lesson 2 to review which strategy is required for each question.
![Page 3: Solving Polynomial Equations PPT 5.3.2. Factor Polynomial Expressions In the previous lesson, you factored various polynomial expressions. Such as: x](https://reader036.vdocuments.net/reader036/viewer/2022082516/56649d835503460f94a69fa5/html5/thumbnails/3.jpg)
Solving Polynomial Equations
The expressions on the previous slide are now equations:
y = x3 – 2x2 and y = x4 – x3 – 3x2 +3x
To solve these equations, we will be solving for x when y = 0.
![Page 4: Solving Polynomial Equations PPT 5.3.2. Factor Polynomial Expressions In the previous lesson, you factored various polynomial expressions. Such as: x](https://reader036.vdocuments.net/reader036/viewer/2022082516/56649d835503460f94a69fa5/html5/thumbnails/4.jpg)
Solve
y = x3 – 2x2 0 = x3 – 2x2
0 = x2(x – 2)
x2 = 0 or x – 2 = 0 x = 0 x = 2
Therefore, the roots are 0 and 2.
Let y = 0
Common factor
Separate the factors and set them equal to zero.
Solve for x
![Page 5: Solving Polynomial Equations PPT 5.3.2. Factor Polynomial Expressions In the previous lesson, you factored various polynomial expressions. Such as: x](https://reader036.vdocuments.net/reader036/viewer/2022082516/56649d835503460f94a69fa5/html5/thumbnails/5.jpg)
Solve
y = x4 – x3 – 3x2 + 3x 0 = x4 – x3 – 3x2 + 3x 0 = x(x3 – x2 – 3x + 3)
0 =x[x2(x – 1) – 3(x – 1)]0 = x(x – 1)(x2 – 3)
x = 0 or x – 1 = 0 or x2 – 3 =
0 x = 0 x = 1 x =
Therefore, the roots are 0, 1 and ±1.73
Let y = 0
Common factor
Separate the factors and set them equal to zero.
Solve for x
3
Group
![Page 6: Solving Polynomial Equations PPT 5.3.2. Factor Polynomial Expressions In the previous lesson, you factored various polynomial expressions. Such as: x](https://reader036.vdocuments.net/reader036/viewer/2022082516/56649d835503460f94a69fa5/html5/thumbnails/6.jpg)
What are you solving for?
In the last two slides we solved for x when y = 0, which we call the roots. But what are roots?
If you have a graphing calculator follow along with the next few slides to discover what the roots of an equation represent.
![Page 7: Solving Polynomial Equations PPT 5.3.2. Factor Polynomial Expressions In the previous lesson, you factored various polynomial expressions. Such as: x](https://reader036.vdocuments.net/reader036/viewer/2022082516/56649d835503460f94a69fa5/html5/thumbnails/7.jpg)
What are roots?
Press the Y= button on your calculator.
Type x3 – 2x2
![Page 8: Solving Polynomial Equations PPT 5.3.2. Factor Polynomial Expressions In the previous lesson, you factored various polynomial expressions. Such as: x](https://reader036.vdocuments.net/reader036/viewer/2022082516/56649d835503460f94a69fa5/html5/thumbnails/8.jpg)
Press the GRAPH button.
Look at where the graph is crossing the x-axis.
The x-intercepts are 0 and 2.
If you recall, when we solved for the roots of the equation y = x3 – 2x2, we found them to be 0 and 2. Don’t forget, we also put 0 in for y, so it makes sense that the roots would be the
x-intercepts.
![Page 9: Solving Polynomial Equations PPT 5.3.2. Factor Polynomial Expressions In the previous lesson, you factored various polynomial expressions. Such as: x](https://reader036.vdocuments.net/reader036/viewer/2022082516/56649d835503460f94a69fa5/html5/thumbnails/9.jpg)
Use your graphing calculator to graph the other equation we solved,
y = x4 – x3 – 3x2 + 3x
As you would now expect, the roots that we found earlier, 0, 1 and ±1.73, are in fact the x-intercepts of the graph.
![Page 10: Solving Polynomial Equations PPT 5.3.2. Factor Polynomial Expressions In the previous lesson, you factored various polynomial expressions. Such as: x](https://reader036.vdocuments.net/reader036/viewer/2022082516/56649d835503460f94a69fa5/html5/thumbnails/10.jpg)
The Quadratic Formula
02
42
awherea
acbbx
For equations in quadratic form: ax2 + bx + c = 0, we can use the quadratic formula to solve for the roots of the equation.
This equation is normally used when factoring is not an option.
![Page 11: Solving Polynomial Equations PPT 5.3.2. Factor Polynomial Expressions In the previous lesson, you factored various polynomial expressions. Such as: x](https://reader036.vdocuments.net/reader036/viewer/2022082516/56649d835503460f94a69fa5/html5/thumbnails/11.jpg)
Using the Quadratic Formula
Solve the following cubic equation:
y = x3 + 5x2 – 9x
0 = x(x2 + 5x – 9)
x = 0 x2 + 5x – 9 = 0
We can, however, use the quadratic formula.
YES it can – YES it can – common factor.common factor.
Can this equation Can this equation be factored?be factored?
We still need to solve for x We still need to solve for x here. Can this equation be here. Can this equation be factored?factored?
No. There are no two No. There are no two integers that will multiply integers that will multiply to -9 and add to 5.to -9 and add to 5.a = 1
b = 5
c = -9
41.1,41.62
615
)1)(2(
)9)(1(4)5()5( 2
x
x
x
Therefore, the roots are 0, 6.41 and -1.41.
Remember, the root 0 came from an earlier step.