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Solving%20Polynomial%20Equations

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Solve for x. Group. What are you solving for? ... quadratic formula to solve for the roots ... We still need to solve for x here. Can this equation be factored? ... – PowerPoint PPT presentation

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Title: 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.
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