Solve cubic, quartic, and higher-degree polynomial inequalities with a full sign analysis chart and graph — handles repeated roots automatically.
Why this page handles polynomial sign problems
Polynomial Solver
Built for cubic, quartic, and higher-degree inequalities with factored-form input, dynamic sign charts, root multiplicity detection, and interval notation.
Supported Input Styles
x^3-x>0 factors into x(x-1)(x+1) — three distinct roots, four intervals.x^3-4x^2+4x<=0 has a repeated root at x=2 — the sign does not flip there.x^4-5x^2+4<0 is a degree-4 polynomial — factor as (x^2-1)(x^2-4) to find four roots.(x-1)^2*(x+2)>0 is already in factored form — multiplicity is read directly.-x^3+3x^2-3x+1<0 has 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 = -2 (multiplicity 1), x = 1 (multiplicity 1), x = 4 (multiplicity 1). The sign chart gives (-2, 1) \cup (4, \infty).
Standard form
Factored form
Roots
x = -2 (m 1), x = 1 (m 1), x = 4 (m 1)
Solution
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.
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Step 2
Move all terms to one side
Rewrite so the right side is 0. This makes sign analysis possible.
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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.
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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.
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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.
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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.
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Step 7
Write the solution
Keep only the intervals where the sign matches the inequality symbol, then write the result in interval notation.
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Sign Chart
The roots split the number line into dynamic intervals. One test value per interval gives the sign pattern.
| Row | (-∞, -2) | (-2, 1) | (1, 4) | (4, +∞) |
|---|---|---|---|---|
| Test value | ||||
Factor: x + 2 | - | + | + | + |
Factor: x - 1 | - | - | + | + |
Factor: x - 4 | - | - | - | + |
Overall sign p(x) | - | + | - | + |
Keep? Matches inequality | ✗ | ✓ | ✗ | ✓ |
Calculator Types
Solve one-variable linear inequalities with steps, interval notation, and a clean number-line graph.
Solve quadratic inequalities with sign analysis, roots, interval notation, and a number-line graph.
Solve cubic, quartic, and higher-degree inequalities with sign charts, multiplicity detection, and interval notation.
Solve rational inequalities with steps, excluded values, sign charts, and interval notation.
Solve absolute value inequalities with AND/OR steps, negative-side special cases, two-absolute-value sign charts, and interval notation.
Solve AND/OR compound inequalities with intersection, union, number-line steps, and interval notation.
Graph linear, quadratic, and absolute value inequalities in two variables with shading, dashed or solid boundaries, and system overlap tools.
Zone 4
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 exact rational-root factoring cannot finish the job, the calculator switches to numerical real-root detection and labels those roots as approximate. For degree 5 and above, results may be shown as decimal approximations because there is no general algebraic formula for all quintic and higher-degree roots. 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.
Enter a cubic, quartic, or higher-degree polynomial inequality such as (x-1)(x+2)(x-4) > 0.
Move everything to one side and find the real roots, using exact rational-root factoring first and approximate numerical roots when needed.
Sort the roots, record each multiplicity, and split the number line into one interval between every neighboring pair of roots.
Test one sample point per interval, then mark whether each interval satisfies the requested >, <, >=, or <= relation.
Convert the kept intervals into interval notation and check whether any repeated root is included or excluded by the endpoint rule.
Multiplicity Flip Visualizer
For (x-1)(x-2)^2(x-3) > 0, the root x = 2 has even multiplicity, so the curve touches the axis and returns to the same side. The roots x = 1 and x = 3 have odd multiplicity, so the curve crosses the axis and the sign changes there.
Selected root
Multiplicity 2 is even.
The graph touches the x-axis and bounces back.
The sign stays NEGATIVE on both sides.
Endpoint rule is separate from flip behavior
For > or <, roots are excluded because p(x)=0. For ≥ or ≤, roots are included because equality is allowed. Even multiplicity only tells you whether the sign changes across that root.
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. |
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-testing method you learned for quadratics — just with more roots to track. This is the same sign chart method used for rational inequalities, without undefined denominator points. For a deeper shared explainer, see the sign chart method guide.
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.
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.
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.
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.
These examples cover the three search cases this page is designed for: a cubic with all simple roots, an even repeated root, and a quartic that must be factored before the sign chart is reliable.
(x-1)(x+2)(x-4) > 0
The roots -2, 1, and 4 cut the number line into four intervals. Each root has multiplicity 1, so the sign flips at every boundary.
Answer
x ∈ (-2, 1) ∪ (4, +∞)
(x-1)(x-2)^2(x-3) > 0
x = 2 has multiplicity 2, so the sign does not flip there; x = 1 and x = 3 are crossing roots.
Answer
x ∈ (-∞, 1) ∪ (3, +∞)
x^4 - 5x^2 + 4 < 0
This quartic factors as (x^2-1)(x^2-4), then as (x-1)(x+1)(x-2)(x+2), giving four simple roots.
Answer
x ∈ (-2, -1) ∪ (1, 2)
The repeated-root example (x-1)(x-2)^2(x-3) > 0 shows the page's core rule: the sign stays negative across x = 2 because the multiplicity is even, while x = 1 and x = 3 are crossing roots. The positive solution intervals are (-∞, 1) and (3, ∞).
Sign Chart
The roots split the number line into dynamic intervals. One test value per interval gives the sign pattern.
| Row | (-∞, 1) | (1, 2) | (2, 3) | (3, +∞) |
|---|---|---|---|---|
| Test value | ||||
Factor: x - 1 | - | + | + | + |
Factor: (x - 2)^2 | + | +(touch) | +(touch) | + |
Factor: x - 3 | - | - | - | + |
Overall sign p(x) | + | - | - | + |
Keep? Matches inequality | ✓ | ✗ | ✗ | ✓ |
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.
Number Line
Touch roots are labeled separately because the sign does not flip as the graph passes that x-value.
Move every term to one side so the inequality compares a cubic polynomial with 0. Factor the cubic if possible, list the real roots in order, split the number line into intervals, test one point in each interval, and keep the intervals whose sign satisfies the inequality.
A sign analysis chart lists the intervals created by the real roots, one test point from each interval, the sign of the polynomial at that point, and whether that interval satisfies the inequality. It prevents skipped intervals when a polynomial has three, four, or more roots.
Multiplicity is the exponent on a repeated factor such as (x - r)^k. Odd multiplicity makes the graph cross the x-axis, so the sign flips. Even multiplicity makes the graph touch the x-axis and bounce back, so the sign stays the same on both sides.
A degree-n polynomial can have at most n distinct real roots. If it has k distinct real roots, those roots split the number line into k+1 sign-test intervals. Repeated roots count once as boundaries, but their multiplicity still controls whether the sign flips.
Yes, but you still need the real roots. If exact factoring does not work, you can use graphing or numerical methods to approximate the x-intercepts, then build the same sign chart from those approximate roots.
Graph the polynomial and find where it crosses or touches the x-axis. The x-intercepts split the number line into intervals. For p(x) > 0, keep the parts of the graph above the x-axis; for p(x) < 0, keep the parts below it. Then confirm with a sign chart.
The sign changes only at roots with odd multiplicity. At an even-multiplicity root, the factor is squared, fourth-powered, and so on, so it does not switch from positive to negative as x passes through the root.
A polynomial equation asks where p(x) equals zero. A polynomial inequality asks where p(x) is positive or negative, so roots are only boundary points; the final answer is usually one or more intervals.
Keep Exploring
Only need quadratic sign testing? Start with the quadratic inequality calculator before moving to higher-degree sign charts.
Convert the final polynomial inequality answer into clean interval notation.
Rational inequalities use a similar sign chart method — see how it applies to fractions.
Radical inequalities add domain and extraneous-root checks before interval notation.