Polynomial Inequalities

Introduction

In AP Precalculus, polynomial inequalities extend the study of polynomial equations by determining where a polynomial expression is positive, negative, or zero.
Instead of finding only the zeros, we analyze intervals of solutions.

Polynomial inequalities are important for:

• Modeling constraints in real-world problems

• Understanding function behavior

• Analyzing graphs and sign changes

• Preparing for calculus concepts such as increasing/decreasing behavior

What Is a Polynomial Inequality?

A polynomial inequality has the form:

$$f(x)>0 , f(x)\ge0 , f(x)<0 , f(x)\le0$$

Where f(x) is a polynomial expression.

Examples

$$x^3-4x\ge0 , 2x^2-5x-3<0$$

Strategy for Solving Polynomial Inequalities

To solve a polynomial inequality:

1. Move all terms to one side

2. Factor the polynomial completely

3. Find the zeros (critical points)

4. Divide the number line into intervals

5. Test the sign of the polynomial in each interval

6. Select intervals that satisfy the inequality

Step 1: Write in Standard Form

Example:

$$x^3-4x\ge0$$

Already in standard form.

Step 2: Factor the Polynomial

$$x^3-4x=x(x-2)(x+2)$$

Step 3: Find Critical Points

Set each factor equal to zero:

$$x=-2 , 0 , 2$$

These points divide the number line into intervals.

Step 4: Sign Analysis (Sign Chart)

Step 5: Choose Correct Intervals

Original inequality:

$$x^3-4x\ge0$$

Solution:

$$[-2,0]\cup[2,\infty)$$

Include the zeros because of “greater than or equal to”.

Solving Polynomial Inequalities Using Graphs

Polynomial inequalities can also be solved by analyzing the graph of $y=f(x)$.

• f(x)>0: graph is above the x-axis

• f(x)<0: graph is below the x-axis

• Zeros are x-intercepts

This method emphasizes graph interpretation, a key AP Precalculus skill.

Effect of Multiplicity on Inequalities

Even Multiplicity

• Graph touches the x-axis

• Sign does not change

Example:

(x−1)2(x+2)≥0

Odd Multiplicity

• Graph crosses the x-axis

• Sign changes

Example:

$$(x+1)^3(x-2)<0$$

Multiplicity helps predict sign changes without test points.

Polynomial Inequalities with Leading Coefficient

End behavior helps predict signs for extreme values of $x$.

Example:

$$-2x^4+x^2-1\le0$$

Even degree with negative leading coefficient:

• Negative for large ∣x∣

Writing Solutions in Interval Notation