next, we use the end behavior of the function and mark the

1
Warm-up

• Graph
• f(x) (x 3)2(x 4)(2x 1)
• For what values of x is the graph of f(x) below
  the x-axis.

2
Solving Polynomial Inequalities
2.8A

• constructing a sign chart

3
Lesson Objectives

• You should be able to synthesize your knowledge
  of end-behavior, finding x-intercepts, finding
  vertical asymptotes, and finding the y-intercept
  of polynomial functions in order to solve
  polynomial inequalities.

4
Polynomial Inequalities

• have the form
• f(x) gt 0
• f(x) lt 0
• f(x) ? 0
• f(x) ? 0
• f(x) ? 0

5
Polynomial Inequalities

• Solving f(x) gt 0 means finding the x-values for
  which the functions graph is strictly above the
  x-axis.
• Solving f(x) lt 0 means finding the x-values for
  which the graph is strictly below the x-axis.

6
Multiplicity Reviewed
When a function is written as a product of linear
factors

• Even multiplicity indicates no sign change (a
  touch go).
• Odd multiplicity indicates a sign change (graph
  crosses the x-axis).

7
Example 1

• Let
• Find
• f(x) gt 0

8
Example 1 (continued)

• Since the polynomial function is already written
  as a product of linear factors, we can easily
  find the real zeros and note their multiplicity.

9
Example 1 (and continued)
First, we mark the zeros and their multiplicity
on an x-axis. This drawing will become a sign
chart.
10
Example 1 (and continued)
Next, we use the end behavior of the function and
mark the signs on the chart.
11
Example 1 (and continued)

-
Finally, working from one end to the other, we
mark the sign of the function between each zero,
paying careful attention to multiplicity.
12
Example 1 (and finished)

-
Our solution (-8, -2) U (-2, 1/2) U (3, 8)
13
Examples

• Put away your graphing calculator and use your
  knowledge of end behavior, finding x-intercepts
  (with their respective multiplicities), and
  finding the y-intercept, to solve the following
  inequalities.

14
Example 2

• Solve
• 2(x - 2)3(x 3)2 gt 0

15
Example 3

• Solve
• -(x 2)4(x 1)(2x2 x 1) gt 0

16
Example 4

• Solve
• 3(x 2)2(x 4)3(-x2 - 2) lt 0

17
Rational Polynomial Inequalities

• Real zeros of the numerator are x-intercepts.
• Real zeros of the denominator are
  discontinuities.
• The multiplicity of a zero indicates whether or
  not the graphs y-value will change signs at that
  zero (regardless of whether a zero of the
  numerator or denominator).

18
Zeros of the Denominator

• Remember that you can not include discontinuities
  in an interval.
• Therefore, any interval that ends or begins at a
  discontinuity will use parentheses, ( or ),
  to indicate that the endpoint is not to be
  included in the interval.

19
Example 5