Calculator Input
Enter polynomial coefficients from highest power to constant. Use integers only, like 2, -3, -11, 6.
Formula Used
Polynomial form: f(x) = anxn + an-1xn-1 + ... + a1x + a0
Rational Root Theorem: Possible rational roots are ± p/q.
Condition: p divides the constant term, and q divides the leading coefficient.
Root test: A candidate is a true rational root when f(p/q) = 0.
The graph plots the polynomial over your selected x-range. The table tests every rational candidate exactly, then the graph helps visualize crossings and end behavior.
How to Use This Calculator
- Enter coefficients in descending power order.
- Use commas between values only.
- Choose the graph minimum and maximum x-values.
- Adjust graph points for smoother curves.
- Set scan intervals for stronger visible root estimates.
- Press Calculate and Graph.
- Review possible roots, confirmed rational roots, and the table.
- Download results as CSV or PDF when needed.
Example Data Table
| Example Polynomial | Possible Rational Roots | Confirmed Rational Roots | Notes |
|---|---|---|---|
| x^3 - 6x^2 + 11x - 6 | ±1, ±2, ±3, ±6 | 1, 2, 3 | Classic polynomial with three rational roots. |
| 2x^3 - 3x^2 - 11x + 6 | ±1, ±2, ±3, ±6, ±1/2, ±3/2 | -2, 1/2, 3 | Includes fractional rational roots. |
| x^4 - 5x^2 + 4 | ±1, ±2, ±4 | -2, -1, 1, 2 | Symmetric equation with four rational roots. |
Frequently Asked Questions
1. What does the Rational Root Theorem do?
It lists every possible rational root of a polynomial with integer coefficients. It does not guarantee every candidate is a root, so each candidate must still be tested.
2. Why must I enter integer coefficients?
The theorem is based on integer divisibility. Decimal coefficients should be scaled first, or rewritten as integers, before using this method accurately.
3. What if the constant term is zero?
Then x = 0 is a root. The calculator detects that case and includes zero among tested roots.
4. Why are some real roots not rational?
Some polynomials have irrational roots like √2 or cube roots. The theorem only targets rational candidates, while the graph helps you notice other real roots.
5. What does the exact evaluation column show?
It shows the polynomial value at each candidate. A result of zero confirms the candidate is a true rational root.
6. Why might the graph miss a repeated root?
Repeated roots can touch the x-axis without crossing it. That makes sign-change scans harder, so the exact candidate table remains the stronger proof.
7. When should I increase graph points?
Increase graph points when the curve looks rough, compressed, or too angular. More points usually produce a smoother plotted polynomial.
8. What do the CSV and PDF exports include?
They include the main summary values and tested candidates. CSV is better for spreadsheets, while PDF is better for printing or sharing.