Rational Zeros Calculator

Rational Zeros Calculator

Use the rational root theorem to list the possible rational zeros of a polynomial.

Last updated: June 2026 | By Patchworkr Team

Calculation steps
Factors of |p| = 1, 2, 3, 6
Factors of |q| = 1, 2
Combine each factor of p with each factor of q as +/- p/q.
Possible rational zeros
6 possible rational zeros
+/- 1+/- 1/2+/- 2+/- 3+/- 3/2+/- 6

What Are Rational Zeros?

The rational root theorem gives the possible rational zeros of a polynomial by combining factors of the constant term with factors of the leading coefficient.

It does not guarantee that each candidate is an actual zero. It only lists the values that must be tested.

How To Find Rational Zeros

  1. List the factors of the constant term p.
  2. List the factors of the leading coefficient q.
  3. Form every fraction p/q and reduce it.
  4. Add a plus or minus sign to each reduced fraction.
possible zeros = +/- p / q

Worked Example

For 2x^3 + 3x^2 - 11x - 6, the possible rational zeros are +/-1, +/-2, +/-3, and +/-6, plus the fractions formed by the leading coefficient.

Factors of 6: 1, 2, 3, 6
Factors of 2: 1, 2
Possible zeros include +/-1, +/-1/2, +/-2, +/-3, +/-3/2, +/-6

Frequently Asked Questions

Do all of the possible zeros have to work?

No. They are candidates, so each one must still be tested in the polynomial.

Can the leading coefficient be zero?

No. If the leading coefficient is zero, the polynomial is no longer in standard form.

What if the constant term is zero?

Then x = 0 is automatically a rational zero.

Are decimal coefficients supported?

No. This tool is built around the integer-based rational root theorem.

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