Polynomial Inequalities

A polynomial inequality compares a polynomial to zero: , . The tempting move — isolate the way you would in Linear Inequalities — does not work, because it would mean dividing by variable factors whose signs you do not know.

The method that does work is the sign chart (also called a sign diagram or the test-point method). A polynomial can only switch between positive and negative at a point where it equals zero. Find those points, split the number line at them, and test one value in each piece.

The Polynomial Inequality Formula

The are the critical values — the roots. Between consecutive roots the product keeps one sign; find that sign with a single test point.

The Sign-Chart Method

1. Move every term to one side so the other side is .
2. Factor the polynomial completely.
3. Find the critical values — set each factor to zero.
4. Draw a number line, mark the critical values, and split it into intervals.
5. Test one point in each interval in the factored form; record or .
6. Select the intervals whose sign matches the inequality. For or , also include the critical values.
7. Write the answer in interval notation.

Worked Example A: A Quadratic Inequality

Solve .

Step 1–2 — one side is already zero; factor:

Step 3 — critical values: and .

Step 4–5 — test the three intervals:

Step 6–7 — we want , so the positive intervals; strict, so exclude the endpoints:

Answer: .

Worked Example B: A Non-Strict Inequality

Solve .

Factor: . Critical values .

We want : the negative interval, and the critical values (where it equals ).

Answer: .

Worked Example C: A Cubic

Solve .

Factor: . Critical values .

We want : the negative intervals, strict.

Answer: .

Worked Example D: A Repeated Factor

Solve .

Critical values (from the squared factor) and .

The sign does not flip at , because always. We want : the positive intervals plus every critical value (both make the product ).

Answer: . Note is included even though it is “inside” the solution — it satisfies as equality.

Why a Polynomial Only Changes Sign at a Root

A polynomial is a continuous, unbroken curve — it never jumps. To get from positive to negative, its graph has to pass through zero, and the only -values where a polynomial equals zero are its roots. Between two consecutive roots there is no zero, so the polynomial keeps whatever sign it has across that entire gap. That is why one test point per interval settles the whole interval, and why the critical values are exactly the roots.

Multiplicity: When the Sign Does Not Flip

At a root that comes from a factor raised to an odd power (multiplicity 1, 3, …), the polynomial does change sign — the graph crosses the axis. At a root from a factor raised to an even power (multiplicity 2, 4, …), the polynomial touches the axis and turns back, so the sign is the same on both sides.

For a non-strict inequality, an even-multiplicity root still gets included in the solution (it makes the product zero, which satisfies or ), even though the sign is the same on both sides of it.

Worked Example E: A Quadratic That Does Not Factor

Solve .

Step 1 — one side is zero; the quadratic does not factor over the integers, so find its roots with the formula:

Numerically, and . These are the critical values.

Step 2 — test the three intervals (any convenient point):

Step 3 — we want : the positive intervals, plus the roots themselves:

Answer: . Irrational critical values are handled exactly like integer ones — only their order on the line matters.

Common Mistakes to Avoid

• Dividing by a variable factor. is not ; dividing by loses and ignores that can be negative. Move everything to one side and factor: .
• Not setting one side to zero first. The sign chart tracks the sign of a single expression versus zero.
• Forgetting the endpoints for / . Non-strict inequalities include every critical value.
• Including endpoints for / . Strict inequalities exclude them.
• Assuming the sign always alternates. It alternates only at simple (odd-multiplicity) roots. At a squared factor it does not flip.
• Picking a test point on a critical value. Test points must be strictly inside an interval.

Where This Shows Up Later

• Rational Inequalities. Same sign-chart method, with the extra step of marking where the denominator is zero as an always-excluded critical value.
• Graphing polynomials. The sign chart is a shortcut for sketching where the graph is above or below the -axis, and multiplicity explains “cross versus touch” at each intercept.
• Quadratic applications with a constraint. “For what dimensions is the area at least 40?” is a polynomial inequality.
• Calculus. Finding where a derivative is positive or negative (to locate increasing/decreasing intervals) is exactly this method.

Practice Problems

Work each problem yourself before opening the answer. Give solutions in interval notation.

Problem 1. Solve .

Problem 2. Solve .

Problem 3. Solve .

Problem 4. Solve .

Problem 5. Solve .

Problem 6. Solve .

Problem 7. Solve .

Problem 8. Solve .

Problem 9. Solve .

Problem 10. Solve .

Problem 11. Solve .

Problem 12. Solve .

Problem 13. Solve .

Problem 14. Solve .

Quick Reference

Factoring uses Factoring Quadratics; the answer format is Interval Notation. The next step, with denominators, is Rational Inequalities. More Algebra lessons are available too.