Rational Zeros Calculator — Rational Root Theorem
Enter the leading coefficient and constant term of a polynomial with integer coefficients to list all possible rational zeros using the Rational Root Theorem. Each candidate is a fraction ±p/q where p divides the constant term and q divides the leading coefficient.
How does this calculator work?
The Rational Root Theorem says any rational zero of a polynomial (in lowest terms p/q) must satisfy: p divides the constant term and q divides the leading coefficient. List all ±(factor of a₀)/(factor of aₙ) combinations and test each by substitution. These are candidates only — not confirmed roots.
Formula
How this is calculated
The Rational Root Theorem (also called the Rational Zero Theorem) states: if a polynomial with integer coefficients has a rational zero expressed in lowest terms as p/q, then p must divide the constant term a₀ and q must divide the leading coefficient aₙ. This gives a finite list of candidates to test by substitution or synthetic division — the theorem does not guarantee these values actually are zeros; it only rules out all other rationals.
To generate the list, find all positive integer factors of |a₀| (call them p-values) and all positive integer factors of |aₙ| (call them q-values). For each pair (p, q) reduce the fraction p/q to lowest terms by dividing by GCD(p, q), then include both +p/q and −p/q. Duplicates (such as 2/2 already reduced to 1/1) are skipped so each candidate appears exactly once.
The candidates can then be tested one by one using the factor theorem: substitute each value into the polynomial and check whether f(candidate) = 0. If a candidate is confirmed as a root, polynomial long division or synthetic division reduces the degree by one and the process continues. This calculator provides the candidate list only — full polynomial evaluation would require all coefficients to be entered.
Frequently asked questions
No — it gives every rational number that could possibly be a zero. Many candidates will not actually be zeros. Each candidate must be tested by substituting it into the polynomial or using synthetic division. Irrational zeros (such as √2) and complex zeros are not covered by the theorem.
All candidates can be tested and found not to be zeros. In that case the polynomial's roots are all irrational or complex and cannot be found with the rational root theorem alone. Numerical methods or the quadratic/cubic formula may be needed.
The theorem only requires the leading coefficient (highest-degree term) and the constant term (zero-degree term) to generate the candidate list. Knowing all intermediate coefficients is necessary only when testing each candidate by substitution.
Also known as
TG we-Calculate Editorial Team. (2026). Rational Zeros Calculator — Rational Root Theorem [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/rational-zeros-calculator
TG we-Calculate Editorial Team. "Rational Zeros Calculator — Rational Root Theorem." TG we-Calculate. 2026. https://we-calculate.com/calculator/rational-zeros-calculator.
TG we-Calculate Editorial Team, "Rational Zeros Calculator — Rational Root Theorem," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/rational-zeros-calculator
@misc{wecalculate_rational_zeros_calculator, title = {Rational Zeros Calculator — Rational Root Theorem}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/rational-zeros-calculator}}, year = {2026}, note = {TG we-Calculate} }
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