Polynomial Inequality Calculator

THE SOLUTION

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Type the three coefficients, then press =.

The squares completed, and the curve drawn

Your polynomial inequality

Type it either way: (x − 1)²(x − 3) > 0 or the expanded x³ − 5x² + 7x − 3 > 0. Powers up to 8 are read. Write a power as x^3. If a factor has no whole-number root this page says so rather than rounding.

Factorised or expanded — either is fine.

Tap a box, then use the keys

Move everything to one side and the roots cut the line into stretches, with one sign each. The part people get wrong is what happens at a repeated root: an odd multiplicity changes the sign, an even one does not — the curve touches and turns back. That is why (x − 1)²(x − 3) > 0 and (x − 1)³(x − 3) > 0 have different answers.

Examples — tap one and it solves

The working, step by step

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This polynomial inequality calculator takes degree 3 and up, finds every real root with its multiplicity, and shows the sign on each stretch.

Type (x − 1)²(x − 3) > 0 and you get x > 3. Change one exponent to a 3 and the answer becomes x < 1 or x > 3. Same roots, different answer.

What the Polynomial Inequality Calculator Handles That a Lower Degree Cannot

Below degree 3 a repeated root has nothing on both sides of it, so the whole answer collapses to a point or to nothing — which is why a quadratic inequality never needs this reasoning. From degree 3 up, a repeated root sits inside a chart with other roots around it, and the multiplicity starts to matter.

Odd Changes the Sign, Even Does Not

Odd multiplicity — 1, 3, 5 — the curve goes through the axis and the sign flips.
Even multiplicity — 2, 4 — the curve reaches the axis and turns back the way it came, so the sign on the far side is the same.

That is the whole difference between the two examples above, and it is what this topic is known for getting wrong. Counting how many times a factor repeats is its own job — the algebraic multiplicity calculator does only that.

Start From the Far Right

Past the largest root, the highest power decides everything, so the sign there is just the sign of the leading coefficient — certain, with nothing to test.

Walk leftwards from there, flipping at each odd root and holding at each even one. That is safer than testing a point in every stretch and hoping the chosen numbers were kind.

When One Number Stands Alone

(x − 1)²(x − 3) ≥ 0 gives {1} together with x ≥ 3.

The 1 is there on its own because the polynomial is zero at a touching root, and a ≥ lets zero in — even though everything around it has the wrong sign. It looks like an error and is not, and only an even multiplicity can produce it.

A Factor With No Real Root Is Invisible

In (x − 1)(x² + 1) the quadratic is positive everywhere. It never reaches the axis, so it adds no root and flips nothing. Only x = 1 matters.

Spotting that early saves work: check the discriminant of any quadratic factor before drawing anything.

When Nothing Factorises

Whole-number and fraction roots come from the divisors of the constant over the divisors of the leading coefficient. If none of them fits and what remains is degree 3 or more, its roots cannot be written down exactly.

This page says so rather than printing a decimal. An approximation is fine for a sketch and wrong as an endpoint, because it moves which numbers fall inside the answer.

Using the Polynomial Inequality Calculator

Type it factorised or expanded, with terms on either side. You get the collected form, every root with its multiplicity, the sign walked from the right, the answer in words and in interval notation, and a drawing where a crossing root is hollow and a touching one is filled.

Polynomial Inequality Calculator FAQ

How do you solve a polynomial inequality?

Collect on one side, factorise, and let the roots cut the number line into stretches. Start from the far right, where the sign is the sign of the leading coefficient, and work leftwards.

Does the sign always change at a root?

No. It changes only at a root of odd multiplicity. At an even one the curve touches the axis and turns back, so the stretch either side carries the same sign.

Why is one number on its own in my answer?

Because a touching root sits in a stretch of the wrong sign, and a non-strict symbol lets it in anyway since the polynomial is zero there. It is a real answer, not a mistake.

What if the polynomial will not factorise?

If no whole number or fraction is a root and what remains is degree three or more, the roots exist but cannot be written down exactly. A decimal is an approximation, not an answer.

Count the multiplicities first — the chart follows from them.