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:
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Modeling constraints in real-world problems
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Understanding function behavior
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Analyzing graphs and sign changes
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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:
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Move all terms to one side
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Factor the polynomial completely
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Find the zeros (critical points)
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Divide the number line into intervals
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Test the sign of the polynomial in each interval
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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)
| Interval | Test Value | Sign of f(x) |
|---|---|---|
| x<-2 | −3 | Negative |
| -2<x<0 | −1 | Positive |
| 0<x<2 | 1 | Negative |
| x>2 | 3 | Positive |
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)y = f(x).
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f(x)>0: graph is above the x-axis
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f(x)<0: graph is below the x-axis
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Zeros are x-intercepts
This method emphasizes graph interpretation, a key AP Precalculus skill.
Effect of Multiplicity on Inequalities
Even Multiplicity
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Graph touches the x-axis
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Sign does not change
Example:
(x−1)2(x+2)≥0
Odd Multiplicity
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Graph crosses the x-axis
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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 xx.
Example:
$$-2x^4+x^2-1\le0$$
Even degree with negative leading coefficient:
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Negative for large ∣x∣
Writing Solutions in Interval Notation
| Inequality | Interval Notation |
|---|---|
| x>2 | (2,∞) |
| x≥−1 | [−1,∞) |
| x<3 | (−∞,3) |
| a≤x≤b | [a,b] |