What degree polynomials does this calculator handle?
The calculator handles degree 3 (cubic), degree 4 (quartic), and higher. For degree 1 (linear) and degree 2 (quadratic), use the dedicated linear and quadratic inequality calculators.
Polynomial Inequality Calculator
Solve cubic, quartic, and higher-degree polynomial inequalities instantly — get step-by-step sign chart analysis, number line graphs, and interval notation. Free, no sign-up required.
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Why this page handles polynomial sign problems
Built for degree-3 and higher inequalities where the number of roots and intervals is not fixed — the sign chart adapts to however many real roots the polynomial has.
Treats even-multiplicity roots (repeated roots) correctly: the sign does not flip across a touch point, and the page labels these explicitly so students do not mistake them for crossing roots.
Pairs the sign chart with a polynomial curve graph and interval notation so the answer is visible algebraically, visually, and symbolically at the same time.
Polynomial Solver
Built for cubic, quartic, and higher-degree inequalities with factored-form input, dynamic sign charts, root multiplicity detection, and interval notation.
Live parse preview
x^3 - x > 0
Supported Input Styles
x^3-x>0factors into x(x-1)(x+1) — three distinct roots, four intervals.x^3-4x^2+4x<=0has a repeated root at x=2 — the sign does not flip there.x^4-5x^2+4<0is a degree-4 polynomial — factor as (x^2-1)(x^2-4) to find four roots.(x-1)^2*(x+2)>0is already in factored form — multiplicity is read directly.-x^3+3x^2-3x+1<0has a triple root at x=1 — odd multiplicity, sign flips once.
Math Keyboard
Tap cubic, quartic, factor, and comparison keys for fast polynomial input.
Result
Degree 3 polynomial with x = -1 (multiplicity 1), x = 0 (multiplicity 1), x = 1 (multiplicity 1). The sign chart gives (-1, 0) \cup (1, \infty).
Rootsx = -1 (m 1), x = 0 (m 1), x = 1 (m 1)
Interval(-1, 0) ∪ (1, ∞)
Degree3
Standard form
Factored form
Roots
x = -1 (m 1), x = 0 (m 1), x = 1 (m 1)
Solution
1
Step 1
Identify the polynomial structure
A polynomial inequality is solved by finding where the polynomial changes sign. The sign can only change at a real root, so the first task is to move everything to one side and compare with zero.
Before
After
2
Step 2
Move all terms to one side
Rewrite so the right side is 0. This makes sign analysis possible.
Before
After
3
Step 3
Factor the polynomial
Factor completely to expose all real roots. Each linear factor (x - r) contributes one root. Each repeated factor (x - r)^k contributes a root of multiplicity k.
Before
After
Even-multiplicity roots are touch points. They can be included for ≥ or ≤, but they do not flip the sign on the chart.
4
Step 4
Find all real roots and their multiplicities
Set each factor equal to zero. Record the multiplicity of each root — it determines whether the sign flips or stays the same at that point.
Before
After
Even-multiplicity roots are touch points. They can be included for ≥ or ≤, but they do not flip the sign on the chart.
5
Step 5
Build the sign chart
The n real roots divide the number line into n+1 intervals. Pick one test value per interval and evaluate the sign of the polynomial.
Before
After
6
Step 6
Apply endpoint rules
Strict inequalities (> or <) exclude all roots. Inclusive inequalities (≥ or ≤) include roots where the polynomial equals zero — but only roots of odd multiplicity actually change the sign.
Before
After
Even-multiplicity roots are touch points. They can be included for ≥ or ≤, but they do not flip the sign on the chart.
7
Step 7
Write the solution
Keep only the intervals where the sign matches the inequality symbol, then write the result in interval notation.
Before
After
Sign Chart
The roots split the number line into dynamic intervals. One test value per interval gives the sign pattern.
x = -1 · multiplicity 1 · crossing
x = 0 · multiplicity 1 · crossing
x = 1 · multiplicity 1 · crossing
| Row | (-∞, -1) | (-1, 0) | (0, 1) | (1, +∞) |
|---|---|---|---|---|
| Test value | ||||
Factor: x + 1 | - | + | + | + |
Factor: x | - | - | + | + |
Factor: x - 1 | - | - | - | + |
Overall sign p(x) | - | + | - | + |
Keep? Matches inequality | ✗ | ✓ | ✗ | ✓ |
MULTIPLICITY NOTE: A root with even multiplicity is a touch point — the polynomial touches zero but does not cross. The sign is the same on both sides of this root.
Use the tabs to move between steps, sign chart, graph, interval notation, and verification.
Calculator Types
Switch to another inequality tool in one tap
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Polynomial
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Compound
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Graphing
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Zone 4
How to Use the Polynomial Inequality Calculator
This polynomial inequality calculator handles cubic, quartic, and higher-degree inequalities where the number of roots and intervals is not fixed in advance. You can enter the polynomial in expanded form such as x^3 - x > 0, in factored form such as (x-1)^2*(x+2) > 0, or in any rearranged form such as x^3 > 4x^2 - 4x. The parser moves everything to one side automatically before beginning the sign analysis.
After the expression is recognized, the page follows the sign chart method that strong precalculus and calculus students use. The Steps tab shows how the polynomial is factored, how the roots are found, and how the sign is read across each interval. The Sign Chart tab makes the interval-by-interval analysis explicit, with one test value per region and a clear label for any root where the sign does not flip. The Polynomial Graph tab shows the curve on the coordinate plane so you can see the crossing and touch behavior directly.
If you are learning the topic for the first time, follow the tabs in order: Steps, Sign Chart, Polynomial Graph, Number Line, Interval Notation. If you are checking homework, go straight to the Sign Chart tab and verify that your test values and sign readings match. Pay particular attention to any root labeled as a touch point — those are the most common source of errors in polynomial inequality work.
If your expression is degree 2, use the quadratic inequality calculator. If the expression has a denominator, use the rational inequality calculator.
01
Enter a polynomial inequality such as x^3 - x > 0 or (x-1)^2*(x+2) > 0 in expanded or factored form.
02
Review the factored form, root list with multiplicities, and sign chart to see which intervals satisfy the inequality.
03
Use the polynomial graph, number line, interval notation, and verify tabs to confirm the answer from multiple representations.
What Is a Polynomial Inequality?
A polynomial inequality compares a polynomial expression with zero. In standard form it looks like p(x) > 0, p(x) < 0, p(x) ≥ 0, or p(x) ≤ 0, where p(x) is a polynomial of degree 2 or higher. This calculator focuses on degree 3 and above — the cases where the sign chart is the only reliable general method because there is no simple "opening direction" shortcut.
The key insight is that a polynomial can only change sign at its real roots. Between any two consecutive roots, the sign stays constant. That is why the sign chart method works: find all real roots, test one value per interval, and keep the intervals whose sign matches the inequality. The number of intervals is always one more than the number of distinct real roots.
Root multiplicity adds one more layer. A root of odd multiplicity is a crossing point — the polynomial changes sign there. A root of even multiplicity is a touch point — the polynomial reaches zero but returns to the same sign. Identifying which roots are crossing points and which are touch points is the step that separates a correct sign chart from a wrong one.
| Comparison point | Quadratic inequality | Polynomial inequality (degree ≥ 3) |
|---|---|---|
| Roots | At most 2 real roots | Up to n real roots (n = degree) |
| Intervals | At most 3 intervals | Up to n+1 intervals |
| Sign shortcut | Opening direction (a > 0 or a < 0) | No shortcut — sign chart required |
| Multiplicity issue | Double root is a special case | Even-multiplicity roots are common and must be identified |
| Graph shape | Parabola | S-curve (cubic), W/M-shape (quartic), etc. |
How to Solve Polynomial Inequalities Step by Step
The sign chart method is the standard approach for polynomial inequalities of any degree. It works because a polynomial's sign can only change at its real roots. Once you know all the roots and their multiplicities, you only need one test value per interval to determine the complete solution set. This is the same sign chart method used for rational inequalities, without undefined denominator points.
1
Factor completely
Move all terms to one side so the right side is zero, then factor the polynomial as completely as possible. If the polynomial does not factor over the rationals, use numerical methods or the rational root theorem to find approximate roots.
2
Identify roots and multiplicities
Each factor (x - r)^k contributes a root at x = r with multiplicity k. Odd multiplicity means the polynomial crosses the x-axis at r. Even multiplicity means the polynomial touches the x-axis at r without crossing. This distinction controls whether the sign flips at each critical point.
3
Build and read the sign chart
List the roots in order from left to right. They divide the real line into intervals. Pick one test value inside each interval and substitute it into the factored polynomial. Record the sign (positive or negative) for each interval. Keep the intervals whose sign matches the inequality symbol.
4
Apply endpoint rules
For strict inequalities (> or <), all roots are excluded — use open endpoints. For inclusive inequalities (≥ or ≤), roots where the polynomial equals zero are included — use closed endpoints. Even-multiplicity roots are included for ≥ and ≤ but they do not change the sign pattern around them.
Polynomial Sign Chart Example
The default example x^3 - x > 0 creates three real roots and four dynamic intervals. The chart keeps only the positive intervals.
Sign Chart
The roots split the number line into dynamic intervals. One test value per interval gives the sign pattern.
x = -1 · multiplicity 1 · crossing
x = 0 · multiplicity 1 · crossing
x = 1 · multiplicity 1 · crossing
| Row | (-∞, -1) | (-1, 0) | (0, 1) | (1, +∞) |
|---|---|---|---|---|
| Test value | ||||
Factor: x + 1 | - | + | + | + |
Factor: x | - | - | + | + |
Factor: x - 1 | - | - | - | + |
Overall sign p(x) | - | + | - | + |
Keep? Matches inequality | ✗ | ✓ | ✗ | ✓ |
MULTIPLICITY NOTE: A root with even multiplicity is a touch point — the polynomial touches zero but does not cross. The sign is the same on both sides of this root.
Polynomial Graph and Number Line
The polynomial graph shows the curve crossing or touching the x-axis at each real root. The number line compresses that behavior into solution intervals and endpoint markers.
The final answer is written in interval notation so the symbolic result matches the visual shading.
Polynomial Graph
The shaded x-intervals show where the polynomial satisfies the inequality. Crossing roots change the sign; touch roots (even multiplicity) do not.
Degree
3
Leading coefficient
1
Roots
-1 (1), 0 (1), 1 (1)
Visible X Range
-3.5 to 3.5
Range: 7Purple curve: p(x)Green overlay: solution intervals
Drag to pan. Use the wheel or buttons to zoom.Number Line
Touch roots are labeled separately because the sign does not flip as the graph passes that x-value.
Open circles exclude roots. Filled circles include roots for ≥ or ≤.Amber roots are touch points; purple roots are crossing points.
Frequently Asked Questions
What is a touch point and why does it matter?
A touch point is a root with even multiplicity. The polynomial reaches zero there but does not cross the x-axis, so the sign is the same on both sides. If you treat a touch point as a crossing point, your sign chart will be wrong for every interval beyond it.
Can I enter the polynomial in factored form?
Yes. Enter (x-1)^2*(x+2)>0 and the calculator will read the factors and multiplicities directly without expanding first.
What if the polynomial has no real roots?
If the polynomial has no real roots, its sign never changes. The solution is either all real numbers or no solution, depending on the leading coefficient and the inequality symbol.
How is this different from the quadratic inequality calculator?
The quadratic calculator uses the parabola's opening direction as a shortcut. This calculator uses a full sign chart because higher-degree polynomials have no equivalent shortcut — the number of roots and intervals varies with the specific polynomial.
What happens at a root of multiplicity 3?
Odd multiplicity (including 3) means the polynomial crosses the x-axis — the sign flips. The behavior looks like an inflection point on the graph: the curve flattens as it crosses but still changes sign.
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