Enter coefficients to find one rational zero. Review candidate factors, checks, graph, and downloadable reports. Designed for practice, teaching, verification, and quick classroom support.
| Polynomial | Coefficient input | Possible rational candidates | One rational zero | Check value |
|---|---|---|---|---|
| x³ - 6x² + 11x - 6 | 1, -6, 11, -6 | ±1, ±2, ±3, ±6 | 1 | f(1) = 0 |
| 2x³ + x² - 8x - 4 | 2, 1, -8, -4 | ±1, ±2, ±4, ±1/2 | -2 | f(-2) = 0 |
| 4x³ - 4x² - x + 1 | 4, -4, -1, 1 | ±1, ±1/2, ±1/4 | 1 | f(1) = 0 |
| x³ + 2x² - x - 2 | 1, 2, -1, -2 | ±1, ±2 | -2 | f(-2) = 0 |
Rational Root Theorem:
If a polynomial is written as:
anxn + an-1xn-1 + ... + a1x + a0 = 0
then any rational zero must have the form:
x = p / q
where p divides the constant term a0 and q divides the leading coefficient an.
Verification method:
The calculator substitutes each candidate into the polynomial using Horner’s method:
b0 = an, bk = an-k + r · bk-1
If the final remainder equals zero, the tested candidate is a rational zero. For decimal or fractional entries, the calculator first scales the polynomial to integer coefficients before generating candidates.
It finds one rational zero of a polynomial, if such a zero exists among the valid Rational Root Theorem candidates. It also verifies the result numerically and visually.
It tests all rational candidates and highlights one rational zero when found. The polynomial may still have other rational, irrational, or complex zeros beyond the first highlighted answer.
Yes. You can use integers, decimals, or simple fractions like 3/4. The calculator scales the polynomial to integer coefficients before building rational candidates.
Some polynomials have no rational zeros at all. Their zeros may be irrational or complex, so every Rational Root Theorem candidate fails the substitution check.
Then x = 0 is immediately a rational zero. The calculator detects that case first and marks it without generating a full candidate list.
The graph is a visual aid. The true decision comes from theorem candidates and substitution checks, while the graph helps you see intercepts and overall curve behavior.
Tolerance is the maximum allowed numerical error when checking whether f(x) is zero. Smaller tolerance gives stricter checking, especially with decimal inputs.
They include the current polynomial summary and the tested candidate table. This makes it easier to save homework steps, class examples, or revision notes.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.