Polynomial Inequality Inputs
Formula Used
General polynomial: P(x) = anxn + an-1xn-1 + ... + a1x + a0
Inequality target: Solve P(x) < 0, P(x) ≤ 0, P(x) > 0, or P(x) ≥ 0.
Method: Find real roots inside the selected search window. These roots split the number line into intervals. Test one value from each interval, evaluate P(x), and keep intervals whose signs satisfy the chosen inequality.
Inclusive cases: For ≤ and ≥, any detected root is also included because P(x) = 0 at that point.
How to Use This Calculator
- Enter coefficients in descending powers, such as 1,-6,11,-6.
- Select the inequality operator you want to solve.
- Choose a search minimum and maximum wide enough to capture real roots.
- Adjust scan segments for finer numeric root detection.
- Choose your preferred decimal precision.
- Press Solve Inequality to show results above the form.
- Review the solution set, root list, and interval sign table.
- Use the CSV or PDF buttons to export the displayed results.
Example Data Table
| Polynomial | Inequality | Detected Real Roots | Expected Solution Set |
|---|---|---|---|
| x2 - 5x + 6 | ≤ 0 | 2, 3 | [2, 3] |
| x3 - 6x2 + 11x - 6 | ≥ 0 | 1, 2, 3 | [1, 2] ∪ [3, ∞) |
| x4 - 5x2 + 4 | > 0 | -2, -1, 1, 2 | (−∞, -2) ∪ (-1, 1) ∪ (2, ∞) |
Frequently Asked Questions
1. What coefficients format should I use?
Enter numbers from the highest power down to the constant term. For x² − 5x + 6, use 1,-5,6. Separate values with commas only.
2. Does this tool solve exact symbolic inequalities?
It uses numeric root detection and interval testing. Many common polynomials return clean results, but difficult cases may appear as approximations.
3. Why do search minimum and maximum matter?
The calculator scans for real roots inside your chosen window. If a root lies outside that range, the displayed interval solution may be incomplete.
4. What does the scan segments setting do?
More segments mean more sampling intervals and finer root detection. Higher values improve accuracy, especially for closely spaced roots, but increase computation time.
5. Are repeated roots handled?
Yes, the calculator attempts to detect repeated roots numerically. Even-multiplicity roots may touch the axis without changing sign, so numeric settings become more important.
6. When are roots included in the final answer?
Roots are included for ≤ and ≥ because the polynomial equals zero there. They are excluded for strict inequalities, such as < or >.
7. What if the polynomial has no real roots?
Then the sign may stay positive or negative across the scanned real line window. The interval table will show whether the chosen inequality is satisfied.
8. Can I save the output for reports?
Yes. Use the CSV button for table data and the PDF button for a printable summary of the inequality, roots, solution set, and interval checks.