1. Section 4.5: Solving Polynomial Equations
Objectives
• Use what you know about factoring
to solve polynomial equations

2. Concept: Factor Polynomial Expressions
In previous lessons, you factored various polynomial expressions. Such as: 1) x3 – 2x2 = 2) x4 – x3 – 3x2 + 3x = =
Common Factor
Common Factor
x2(x – 2)
x(x3 – x2 – 3x + 3)
x[x2(x – 1) – 3(x – 1)]
Grouping – common factor the first two terms and then the last two terms.
x(x2 – 3)(x – 1)

3. Concept: 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.

4. Concept: Solving Polynomial Equations cont…
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.
1. Let y = 0
2. Common factor
3. Separate the factors and ….set them equal to zero.
4. Solve for x

5. Concept: Solving Polynomial Equations cont…
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 = x = x =
Therefore, the roots are 0, 1, 1.73, and –1.73
1. Let y = 0
2. Common factor GCF
3. Group and factor out GCF
Regroup:… (outside stuff)(inside stuff)
5. Separate the factors and set …..them equal to zero.
6. Solve for x

6. Concept: What are you solving for?
In the last two slides we solved for x when y = 0, which we call the roots. Other names would be: solutions zero’s x-intercepts

7. Concept: What are you solving for? cont…
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. You may write this on your Calculator Skill Sheet

8. Concept: What are you solving for? cont…
1. Press the Y= button on your calculator. 2. Type your equation. Ex: Y1= x3 – 2x2 and Y2=0
Before students continue and graph the equation, have them think about what the equation is going to look like.

9. Concept: What are you solving for? cont…
3. Press the graph button to check.
Look at where the graph is crossing the x-axis.
Notice the graph crosses the x-axis at 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.

10. Concept: 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.
Can this equation be factored?
We still need to solve for x here. Can this equation be factored?
YES it can – common factor.
Remember, the root 0 came from an earlier step.
No. There are no two integers that will multiply to -9 and add to If you need help with this, see your notes from Unit 3.
a = 1
b = 5
c = -9
Therefore, the roots are 0, 6.41 and

11. Homework:
Homework:
IXL
Skill:
A2:K.8