Example Data Table
| Polynomial |
Inequality |
Expected Solution |
| x^2 - 5x + 6 |
P(x) ≤ 0 |
[2, 3] |
| x^2 - 4 |
P(x) > 0 |
(-∞, -2) ∪ (2, ∞) |
| -x^2 + 1 |
P(x) ≥ 0 |
[-1, 1] |
| x^3 - x |
P(x) ≠ 0 |
(-∞, -1) ∪ (-1, 0) ∪ (0, 1) ∪ (1, ∞) |
How to Use This Calculator
- Select the degree of the polynomial.
- Enter coefficients from the highest power to the constant term.
- Choose the relation, such as less than or greater than zero.
- Set the root search range if needed.
- Click the solve button to view intervals, roots, signs, and graph.
- Use CSV or PDF export for reports and homework records.
Understanding Polynomial Inequalities
What the Calculator Does
A polynomial inequality compares a polynomial with zero. It may use greater than, less than, greater than or equal to, or less than or equal to. This calculator studies that comparison by finding important boundary values. Those values are the real roots of the polynomial. Roots split the number line into smaller intervals. Each interval has one stable sign unless another root appears inside it.
Why Roots Matter
Roots are the turning points for the sign test. The polynomial may change from positive to negative at a root. It may also touch the x-axis and keep the same sign. That is common with repeated roots. The calculator checks roots and nearby intervals. It then marks each interval as included or excluded.
Using the Sign Test
The sign test is a reliable algebra method. First, solve P(x) = 0. Next, place the roots on a number line. Then choose one test point in each interval. Substitute that value into the polynomial. A positive or negative result tells whether the interval satisfies the inequality. This calculator automates those steps.
Interpreting the Answer
The answer is shown using interval notation. Parentheses mean an endpoint is not included. Brackets mean an endpoint is included. For strict inequalities, roots are usually excluded. For inequalities with equality, roots are included when they make the polynomial equal to zero. The graph adds a visual check. It shows where the curve sits above or below the x-axis.
Best Practice
Use a wide search range for higher-degree polynomials. Increase samples when roots are close together. Use smaller tolerance for cleaner numerical roots. Always compare the sign table with the graph. This gives a stronger final answer.
FAQs
1. What is a polynomial inequality?
A polynomial inequality compares a polynomial expression with zero or another value. Common signs include greater than, less than, greater than or equal to, and less than or equal to.
2. Why does the calculator find roots first?
Roots are boundary points. They divide the number line into intervals. The polynomial sign stays stable inside each interval, so one test point can represent that full interval.
3. Are endpoints included in the answer?
Endpoints are included when the inequality has equality, such as ≥ or ≤. They are excluded for strict relations like > or <.
4. What does the sign table show?
The sign table shows each interval, a test point, the polynomial value, its sign, and whether that interval belongs to the solution set.
5. Can this calculator solve higher-degree inequalities?
Yes. It supports degrees up to six. It uses numerical root detection, interval testing, and graphing to build the final solution.
6. What if a root is missing?
Increase the search range and sample count. Some roots may sit outside the selected range or may be very close to another root.
7. Why does tolerance matter?
Tolerance controls how close a value must be to zero. Smaller tolerance gives stricter roots. Larger tolerance may catch near roots more easily.
8. Can I export the result?
Yes. You can download a CSV file or create a PDF report. Both options help save roots, intervals, signs, and the final solution.