Rational Root Theorem Calculator

Discover possible rational zeros instantly. Test each candidate with exact evaluation. Visualize polynomial behavior and confirm roots with confidence.

Enter integer coefficients from highest degree to constant term. The calculator lists theorem candidates, checks each value, shows real rational roots, and draws the polynomial.

Calculator Form

Example Data Table

Polynomial	Constant Factors	Leading Factors	Possible Rational Roots	Actual Rational Roots
x^2 - 5x + 6	1, 2, 3, 6	1	±1, ±2, ±3, ±6	2, 3
2x^3 - 3x^2 - 11x + 6	1, 2, 3, 6	1, 2	±1, ±2, ±3, ±6, ±1/2, ±3/2	-2, 1/2, 3
3x^2 + x - 2	1, 2	1, 3	±1, ±2, ±1/3, ±2/3	-1, 2/3

Formula Used

For a polynomial with integer coefficients, if r = p/q is a rational root in lowest terms, then p divides the constant term and q divides the leading coefficient.

Rational Root Theorem:
If P(x) = a_nxⁿ + ... + a₁x + a₀,
then possible rational roots are:

x = ± p / q

where p | a₀ and q | a_n

After generating candidates, the calculator substitutes each value into the polynomial. When P(r) equals zero, that candidate is a true rational root.

How to Use This Calculator

Enter polynomial coefficients from highest degree to constant.
Separate every coefficient with a comma.
Choose a graph range for x-values.
Select the number of graph sample points.
Click the calculate button.
Review the candidate list from the theorem.
Check which candidates produce zero exactly.
Use the graph to visually confirm root locations.
Download the testing table as CSV or PDF.

Frequently Asked Questions

1. What does this calculator find?

It finds possible rational roots for a polynomial, tests each candidate, identifies exact rational zeros, and plots the polynomial on a graph.

2. Why must coefficients be integers?

The rational root theorem depends on integer divisibility. Integer coefficients allow the numerator and denominator rules to work correctly.

3. Does every candidate become a root?

No. The theorem gives only possible rational roots. Each candidate must still be tested by substitution or division.

4. Can irrational roots appear?

Yes. A polynomial may contain irrational or complex roots. This tool focuses on rational candidates produced by the theorem.

5. What if the constant term is zero?

Then zero is a root. The calculator includes zero and continues checking the reduced polynomial when appropriate.

6. Why show the quotient after division?

The quotient helps verify factorization. Once a root is confirmed, the remaining polynomial reveals other roots more easily.

7. What does the graph add?

The graph shows where the curve crosses or touches the x-axis. That visual check supports the exact symbolic test results.

8. Can I use decimals in coefficients?

This version expects integers only. Decimals break the standard divisibility rules used by the rational root theorem.