➗

Rational Root Theorem Calculator (Possible and Actual Rational Zeros)

List every possible rational root ±p/q of a polynomial with integer coefficients, test each candidate, and get the actual rational zeros with their multiplicity and the factored form.

Polynomial coefficients (integers)

Loading calculator…

How it works

If P(x) = aₙxⁿ + … + a₀ has integer coefficients, every rational root p/q in lowest terms has p dividing a₀ and q dividing aₙ · candidates = ±(factors of a₀)/(factors of aₙ)

The rational root (rational zero) theorem narrows the search for rational solutions of a polynomial equation with integer coefficients to a finite list: ± every factor of the constant term divided by every factor of the leading coefficient. This calculator builds that list, tests every candidate exactly (no rounding), and reports the true rational roots. It then divides each root out repeatedly to find its multiplicity and writes the polynomial in factored form; anything left over has no rational roots (its roots are irrational or complex).

Worked example

For 2x³ − 3x² − 11x + 6: p divides 6 (1, 2, 3, 6) and q divides 2 (1, 2), so the candidates are ±1/2, ±1, ±3/2, ±2, ±3 and ±6 — 12 in total. Testing them, P(−2) = 0, P(1/2) = 0 and P(3) = 0, so the rational roots are −2, 1/2, 3 and the factored form is (x + 2)(2x − 1)(x − 3).

Sources: Rational root theorem — Wikipedia · OpenStax College Algebra 2e — Zeros of polynomial functions

Frequently asked questions

Does the theorem find all roots?

Only the rational ones. A polynomial such as x² − 2 has candidates ±1, ±2 and none works, because its roots ±√2 are irrational. Once you find a rational root, divide it out (synthetic division) and solve the smaller polynomial with other methods.

What if the constant term is 0?

Then x = 0 is a root. Factor out x (as many times as needed) and apply the theorem to what remains, which is what this calculator does automatically.

What if my coefficients are fractions?

Multiply the whole polynomial by the least common denominator first — this does not change the roots — and enter the resulting whole numbers.

Why is a root listed with ×2?

It is a repeated root (multiplicity 2): the factor appears squared, as in (x − 1)² — the graph touches the x-axis there instead of crossing it.

Related calculators

Synthetic Division Calculator

Divide a polynomial by x − r with synthetic division: see the full tableau, the quotient, the remainder, the remainder theorem check and whether x − r is a factor.

Cubic Equation Solver

Solve ax³ + bx² + cx + d = 0 for its real roots using the trigonometric and Cardano methods, with near-integer roots snapped cleanly.

Polynomial Calculator

Add, subtract, multiply or divide two polynomials. Type them as expressions like x^2 − 3x + 2 or (x − 1)(x + 2), or as coefficient lists, and get the simplified result, quotient and remainder, degree, coefficients, a value check and the long-division steps.

Quadratic Equation Solver

Solve ax² + bx + c = 0 with the quadratic formula — real or complex roots, plus the discriminant.

Factoring Trinomials Calculator

Factor any trinomial ax² + bx + c over the integers with the ac (grouping) method, step by step: GCF, the two numbers, splitting the middle term and the final factored form.

This calculator is for educational purposes. Double-check important results before acting on them.