Polynomial & Rational Functions

Polynomial Inequalities

Solving inequalities involving polynomial expressions using sign analysis.

Advanced20 min lesson3 min readUpdated August 12, 2026Author not yet attributed

Prerequisites

Analyzing Polynomial Graphs

Where a Polynomial's Sign Can — and Can't — Change

A polynomial's graph is one continuous, unbroken curve — no jumps, no gaps. Before reading on, think about what that fact implies: could a continuous curve switch from positive to negative values without passing through zero somewhere in between?

It couldn't — a continuous curve moving from a positive value to a negative value has to cross the x-axis somewhere along the way. That means a polynomial's sign can only change at its zeros, which is exactly why the zeros split the number line into intervals where the sign stays constant throughout each entire interval.

Worked Example — Solving a Quadratic Inequality by Sign Analysis

Solve x² − x − 6 > 0. Factor: (x − 3)(x + 2) > 0, with zeros at x = −2 and x = 3, splitting the line into three intervals. Test x = −3 (left of −2): (−3−3)(−3+2) = (−6)(−1) = 6 > 0, true. Test x = 0 (between −2 and 3): (0−3)(0+2) = (−3)(2) = −6, false. Test x = 4 (right of 3): (4−3)(4+2) = (1)(6) = 6 > 0, true. The solution is x < −2 or x > 3, written (−∞, −2) ∪ (3, ∞).

Worked Example — Handling a Zero of Even Multiplicity

Solve (x − 1)²(x + 2) ≤ 0. The zero at x = 1 has multiplicity 2 (even — the sign won't flip there), and the zero at x = −2 has multiplicity 1 (odd — the sign will flip there). Testing intervals: for x < −2 (e.g. x = −3), the expression is negative — satisfies ≤ 0. Between −2 and 1 (e.g. x = 0), the expression is positive — doesn't satisfy. For x > 1 (e.g. x = 2), still positive (since the even-multiplicity zero at x = 1 didn't flip the sign) — doesn't satisfy. The zeros themselves, x = −2 and x = 1, both give exactly 0, satisfying the ≤. The solution is (−∞, −2] ∪ {1}.

Tip

Sketch a quick number line with every zero marked, and test just one point in each resulting interval — a full sign chart avoids having to test the original inequality at every possible value.

Common Mistakes

Assuming every zero flips the polynomial's sign, without checking whether that zero's multiplicity is odd or even.
Only an odd-multiplicity zero actually flips the sign — an even-multiplicity zero touches zero but the sign stays the same on both sides, exactly like it doesn't cross the x-axis when graphed.
Using the wrong type of bracket in interval notation, such as using a closed bracket for a strict inequality (< or >).
A strict inequality (< or >) excludes its boundary zeros, using parentheses; a non-strict inequality (≤ or ≥) includes them, using square brackets.

Key Takeaways

A polynomial's sign can only change at its zeros, since its graph is one continuous, unbroken curve.
Testing one point in each interval between consecutive zeros determines the polynomial's sign throughout that whole interval.
An even-multiplicity zero doesn't flip the sign, even though the expression still equals 0 there.

Summary

Polynomial inequalities use the same zero-and-sign structure that shapes a polynomial's graph. The next lesson extends this graph-analysis toolkit to rational functions, adding one more kind of asymptote.

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