Rational root theorem

A zero leading coefficient or a zero constant term is rejected with invalid_input so callers never receive a silently truncated or infinite candidate set.

What the rational root theorem states and how to encode your polynomial

The degree is one less than the length of a well-formed array and must be at least one, so you always send both a leading term and a constant term. Middle coefficients may be zero; that simply means a missing power of x and does not change the theorem. Coefficients must be integers or near-integers that snap cleanly, because the classical statement of the theorem is for integer coefficient rings. Write a univariate polynomial with integer coefficients as a_n x to the n plus a_{n-1} x to the n-1 down to the constant term a_0. The input array is exactly those integers in that order: the first entry is the leading coefficient a_n and the last is the constant term a_0. The theorem says that every rational root written in lowest terms as p over q has p dividing a_0 and q dividing a_n. What must not be zero under this capability is the leading coefficient and the constant term: if either vanishes the request fails with invalid_input rather than guessing a lower degree or treating zero as a free root outside the classical finite list.

How candidates are generated: factors, signs, and reduced fractions

The response echoes the cleaned coefficients, the degree, a human-readable polynomial string, the leading and constant coefficients, the positive factor lists, the numeric candidates, parallel fraction strings such as 1/2 or -3, structured candidate_details with p, q, value, and fraction, the candidate_count, and a method label fixed to rational_root_theorem. Failed validation—missing array, non-integer entries, zero leading term, or zero constant term—raises invalid_input and is not charged.

Calling the API, reading the response, pricing, and how this fits real workflows

Use the free in-browser tool for one-off homework checks, lecture demos, and quick sanity tests without installing Maxima, SymPy, or a local CAS. Degree is capped and coefficient magnitudes are bounded for isolate safety; very large integers with many divisors remain finite but may produce long candidate lists, so prefer classroom-scale inputs. When you need actual real roots rather than candidates alone, pair this list with synthetic division or call a root-finding capability. Send a JSON body with the required coefficients array of integers, highest degree first; the alias coeffs is accepted for convenience. On success use candidates or candidates_as_fractions for display, and candidate_details when you need exact numerators and denominators for synthetic division loops or step-by-step explanations. Use the API when a tutoring platform, coding-assessment sandbox, curriculum generator, or data pipeline needs a stable contract under prepaid KIT balance at $0.002 per successful item. Source inputs follow the same retention rules as other KIT parse items, and validation failures never consume balance after the engine rejects the request.