# Solving Polynomial Inequalities

How to solve polynomial inequalities problems: examples and their solutions.

## Example 1

Use the synthetic division

to factor the left side of the inequality.

(*x* - 1)(*x* - 2)(*x* + 6) > 0

Factor theorem

The zeros are 1, 2, and -6.

Mark the zeros on the number line: -6, 1, 2.

Draw empty circles on the zeros.

Graph *y* = (*x* - 1)(*x* - 2)(*x* + 6).

Start from the top right. (brown point)

(*x* - 1), (*x* - 2), and (*x* + 6) are all odd powers.

So the graph passes through the number line

at *x* = 2, 1, and -6.

Graphing polynomial functions

The left side is greater than 0.

So draw the region on the number line

where the graph is above the number line.

Then -6 < *x* < 1, *x* > 2.

## Example 2

Use the synthetic division

to factor the left side of the inequality.*x*(*x* - 1)^{2}(*x* + 3) ≤ 0

Factor theorem

The zeros are 0, 1, and -3.

Mark the zeros on the number line: -3, 0, 1.

Draw full circles on the zeros.

Graph *y* = *x*(*x* - 1)^{2}(*x* + 3).

Start from the top right. (brown point)

(*x* - 1)^{2} is an even power.

So the graph bounces off the number line

at *x* = 1.*x* and (*x* + 3) are odd powers.

So the graph passes through the number line

at *x* = 0, -3.

Graphing polynomial functions

The left side is lesser than (or equal to) 0.

So draw the region on the number line

where the graph is below the number line.

Then -3 ≤ *x* ≤ 0, *x* = 1.

## Example 3

Factor the left side of the inequality.

Factoring the difference of two cubes

(*x*^{2} + *x* + 1) is always (+).

So divide both sides by (*x*^{2} + *x* + 1).

It won't change the order of the inequality sign,

nor change the answer.

Let's see why (*x*^{2} + *x* + 1) is always (+).

The discriminant *D* is (-).

So *x*^{2} + *x* + 1 = 0 has no real roots.

And *y* = *x*^{2} + *x* + 1 is always above the *x*-axis.

Quadratic function: number of roots

So (*x*^{2} + *x* + 1) is always (+)

and has no real zeros.

Mark the zeros on the number line: 0, 1.

Draw empty circles on the zeros.

Graph *y* = *x*(*x* - 1) on the number line.

The left side is lesser than 0.

So draw the region on the number line

where the graph is below the number line.

Then 0 < *x* < 1.

Solving quadratic inequalities