Rational Zeros Calculator
Find possible rational zeros (Rational Root Theorem), test them, and get the actual rational zeros (with multiplicity) — plus optional synthetic division steps.
Background
If a polynomial has integer coefficients, any rational zero must be of the form p/q, where p divides the constant term and q divides the leading coefficient.
How to use this calculator
- Choose Coefficients or Polynomial text.
- Enter your polynomial with integer coefficients.
- Click Calculate to see possible rational zeros and the actual ones found.
- Turn on Steps to view synthetic division for each root found.
How this calculator works
- Uses the Rational Root Theorem to generate candidates ±p/q.
- Tests each candidate using exact fraction arithmetic (no floating-point rounding).
- When a root is found, it divides the polynomial and repeats to detect multiplicity.
Formula & Equations Used
Rational Root Theorem: If r = p/q is a rational zero (in lowest terms), then p | a₀ and q | aₙ.
Candidate set: R = { ±p/q } for all divisors p of constant term and q of leading coefficient.
Example Problems & Step-by-Step Solutions
Example 1 — Find rational zeros
f(x)=x^3-6x^2+11x-6 has rational zeros 1,2,3 (each once).
Example 2 — A fractional rational root
f(x)=2x^2 + x - 1 has rational zeros x=1/2 and x=-1.
Frequently Asked Questions
Q: Will this find irrational or complex zeros?
This tool finds rational zeros. If the remaining factor has degree ≥ 2, it may still have irrational or complex zeros.
Q: What if my coefficients aren’t integers?
The Rational Root Theorem is guaranteed only for integer coefficients. The calculator will warn you if it detects non-integers.
Q: Why can there be so many candidates?
If the constant term or leading coefficient has many factors, the candidate list grows. Use the candidates table option if you want the full p/q breakdown.
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