How the Rational Zeros Calculator works
This calculator applies the Rational Root Theorem to a cubic polynomial P(x) = ax³ + bx² + cx + d, where a, b, c, and d are integers and a ≠ 0. The theorem narrows down every number that could possibly be a rational zero, then the calculator tests each candidate directly to see which ones actually satisfy P(x) = 0.
The Rational Root Theorem
If a polynomial with integer coefficients has a rational zero p/q (written in lowest terms), then p must divide the constant term d and q must divide the leading coefficient a. This does not guarantee any of those p/q values are actually zeros — it only produces the complete list of candidates worth testing. For example, x³ − 6x² + 11x − 6 has constant term −6 (divisors ±1, ±2, ±3, ±6) and leading coefficient 1 (divisor ±1), so the only possible rational zeros are ±1, ±2, ±3, ±6. Testing each shows that 1, 2, and 3 all work, so this polynomial factors as (x − 1)(x − 2)(x − 3).
Verifying a candidate
A candidate p/q is confirmed by substituting it into P(x). To avoid rounding error, the calculator instead checks the equivalent integer equation a·p³ + b·p²q + c·pq² + d·q³ = 0, which is exact for whole-number coefficients. Every candidate that satisfies this equation is a genuine rational zero.
Finding the remaining zeros
A cubic has exactly three zeros (counting multiplicity and complex values). Once one rational zero r is confirmed, the calculator divides P(x) by (x − r) using synthetic division, leaving a quadratic Ax² + Bx + C. The quadratic formula, x = (−B ± √(B² − 4AC)) / (2A), then gives the remaining two zeros exactly — they may be rational, irrational, or a complex conjugate pair (when B² − 4AC is negative).
Common sources of error
- Non-integer coefficients: the Rational Root Theorem only applies to polynomials with integer coefficients — clear fractions or decimals first by multiplying through.
- Forgetting negative candidates: every positive candidate p/q has a matching negative one, −p/q, that must also be tested.
- Stopping after one zero: a candidate that fails the test is not a zero, but a cubic can have up to three rational zeros — test every candidate, not just the first that looks promising.
When there are no rational zeros
Not every cubic has a rational zero. If none of the p/q candidates satisfy the equation, the polynomial's real zero (a cubic always has at least one, by the Intermediate Value Theorem) is irrational, and the Rational Root Theorem cannot find it directly — a numerical method (like Newton's method) or the general cubic formula is needed instead.