5.1B Solving Polynomial Inequalities

Solving Polynomial Inequalities

Solving polynomial inequalities will help find solutions to polynomial functions

At some point in Calc, you will be expected to determine where the function increases/decreases or where it is positive/negative.

All of this is to be completed without the use of a calculator.

The solutions will be completed by finding the first and second derivatives of a function, but the process will be the same as determining the solutions for quadratic inequalities or quadratic equations.

What does this process involve? (These steps should be familiar to you.)

~ Find the factors of the polynomial equation.

~ Set the factors equal to zero to identify the function's zeros.

~ Use Test Points to determine the behavior (+/-) of the function around these zeros.

Note - There are two possible situations or questions that will indicate how you will use the results of your work.

A. If asked to find the solutions to the inequalities, use the result of #3 to determine the solution set for the inequality.

B. If asked to determine where the function is above or below the x-axis, use the results of #3 to describe this behavior.

As we work through some problems, it will make sense.

Poly Inequalities Graphs

Example 1

The problem below is asking you to find the solution set (A above).

To understand the process, work through this example (don't just read it, do the math)

9(x - 4) - x2(x - 4) < 0 Given the inequality find the solution set.

Find the common binomial factors

let z = x - 4 The zeros of the function (at the left) have been determined.

9z - x2z < 0

z(9 - x2) < 0

z(3 - x) (3 + x) < 0

(x - 4) (3 - x) (3 + x) < 0 These are the zeros to help determine the behavior of this function.

Set each factor equal to 0

x - 4 = 0 x = 4

3 - x = 0 x = 3

3 + x = 0 x = -3 These zeros (-3, 3, 4) are plotted on a number line, which are used to determine the behavior of the function.

Plot these Critical Points and include test points

<------------(-3)------------(3)-------------(4)----------> zeros

-4 0 3.5 5 Test Points

Use the test points to determine the behavior of f(x)

Pos Neg Pos Neg Behavior of f(x) (see work below)

<------------(-3)------------(3)-------------(4)----------> zeros

-4 0 3.5 5 Test Points

Work to determine the behavior of the function around the zeros

Test points plug-ins using the following factors (x - 4) (3 - x) (3 + x)

~ When x = -4 (-4 - 4) (3 - -4) (3 + -4) results in

(neg) * (pos) * (neg) equals positive - the first region is Pos

~ When x = 0 (0 - 4) (3 - 0) (3 + 0) results in

(neg) * (pos) * (pos) equals negative - the second region is Neg

~ When x = 3.5 (3.5 - 4) (3 - 3.5) (3 + 3.5) results in

(neg) * (neg) * (pos) equals positive - the third region is Pos

~ When x = 5 (5 - 4) (3 - 5) (3 + 5) results in

(pos) * (neg) * (pos) equals negative - the fourth region is Neg

Because this is a "<" inequality, the solution set must have negative values.

This occurs in the second and fourth regions. We must use the zeros for the intervals.

The Solution Set will be (-3,3) U (4, ∞). Notice the endpoints are NOT included because the inequality is less than.

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