Solving%20Polynomial%20Equations

1
Solving Polynomial Equations

• PPT 5.3.2

2
Factor Polynomial Expressions

• In the previous lesson, you factored various
  polynomial expressions.
• Such as
• x3 2x2
• x4 x3 3x2 3x

Grouping common factor the first two terms and
then the last two terms.
Refer to 5.2.2 in Lesson 2 to review which
strategy is required for each question.
Common Factor
x2(x 2)
x(x3 x2 3x 3)
xx2(x 1) 3(x 1)
Common Factor
x(x2 3)(x 1)
3
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
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
5
Solve
Let y 0

• y x4 x3 3x2 3x
• 0 x4 x3 3x2 3x
• 0 x(x3 x2 3x 3)
• 0 xx2(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

Common factor
Group
Separate the factors and set them equal to zero.
Solve for x
6
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.

7
What are roots?

• Press the Y button on your calculator.
• Type x3 2x2

8

• 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. Dont forget, we also put 0 in for y, so
  it makes sense that the roots would be the
• x-intercepts.

9

• 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.

10
The Quadratic Formula

• 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.

11
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.
No. There are no two integers that will multiply
to -9 and add to 5.
a 1 b 5 c -9
Therefore, the roots are 0, 6.41 and -1.41.