Rational Zeros Calculator
Apply the Rational Zero Theorem to list possible rational roots of a polynomial.
Enter integer coefficients up to degree 4 and see which test as actual zeros.
The Rational Zero Theorem narrows down the search for rational roots before any solving begins. If a polynomial with integer coefficients has a rational root p/q (in lowest terms), then p must be a factor of the constant term and q must be a factor of the leading coefficient.
For example, in f(x) = 2x^3 - 3x^2 - 11x + 6:
- Factors of 6 (constant): ±1, ±2, ±3, ±6
- Factors of 2 (leading): ±1, ±2
- Possible rational zeros: ±1, ±2, ±3, ±6, ±1/2, ±3/2
Test each candidate by substituting into f(x). If f(p/q) = 0, it is an actual root. This calculator tests all candidates numerically and reports which ones pass (|f(p/q)| < 0.0001 to account for floating-point rounding).
Once you find a rational root r, you can factor out (x - r) using polynomial long division or synthetic division, reducing the degree by one and making the remaining roots easier to find.
The theorem only applies to polynomials with integer coefficients. For polynomials with irrational or decimal coefficients, you would need numerical root-finding methods instead.
The total number of candidates can grow quickly. A polynomial with constant term 12 and leading coefficient 6 has 8 factors each, giving up to 16 fractions to test before simplifying by GCD. This is still far faster than blind guessing or bracketing methods when a rational root exists.
If the list shows no actual zeros, all roots of this polynomial are either irrational or complex.
How we build and check this calculator
This calculator runs entirely in your browser, so the numbers you enter stay on your device. The math behind it is written by hand and tested against worked examples and standard references before the page goes live.
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