Rational Zeros Calculator

Q: Will this find irrational or complex zeros?

This calculator finds rational zeros. If a remaining factor has degree 2 or higher, it may have irrational or complex zeros.

Q: What if my coefficients aren’t integers?

The Rational Root Theorem is guaranteed only for integer coefficients. If non-integers are used, the candidate list may be incomplete.

Q: Why can there be so many candidates?

The candidate set depends on the factors of the constant term and leading coefficient, so it can grow when those numbers have many divisors.

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.

Enter your polynomial

Input mode:

Coefficients (recommended)

Polynomial text

Degree

Enter coefficients from highest power down to constant term.

Coefficients

Example for 2x^3 - 3x^2 - 8x + 12 → [2, -3, -8, 12]

Polynomial preview

Tip: Switch to “Polynomial text” if you prefer typing the expression.

Options:

Show synthetic division steps (for roots found)

Show full candidates table (p/q)

Reduce fractions

Show graph of f(x) (with rational zeros marked)

Result:

No results yet. Enter a polynomial and click Calculate.

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.

No practice sets found.