Rational Root Test Calculator

This rational root test calculator lists the possible rational roots of a polynomial with integer coefficients, using the rational root theorem. Enter the leading coefficient and the constant term, and the calculator returns every candidate of the form plus or minus a factor of the constant divided by a factor of the leading coefficient. The theorem guarantees that any rational root must be on this list, which turns finding roots into a short search: test each candidate in the polynomial and keep the ones that give zero. It does not promise any of them are actual roots, only that no other rational root exists. The method, a worked example and the assumptions are below.

Leading coefficient

Constant term

Number of candidates

Constant and leading

The formula

The rational root theorem says any rational root p/q in lowest terms of an integer polynomial has p dividing the constant term and q dividing the leading coefficient. The candidates are plus or minus (each factor of the constant) over (each factor of the leading coefficient).

Worked example

For a polynomial with leading coefficient 1 and constant minus 6, the candidates are plus or minus 1, 2, 3 and 6. Testing these in the polynomial finds the actual roots. Enter 1 and -6 to confirm.

Frequently asked questions

What is the rational root theorem?

It says any rational root of an integer polynomial is a factor of the constant term over a factor of the leading coefficient.

Does it find the roots?

It lists the candidates. You still test each one in the polynomial to see which are genuine roots.

What if none work?

Then the polynomial has no rational roots; its real roots are irrational or it has only complex roots.

Who this calculator is for

This calculator is for algebra students and anyone needing a quick, reliable result.

What this calculator assumes

You enter valid numbers.
The standard method is applied.
Results are rounded for display.

Formula and sources

Related calculators

Quadratic Formula Calculator: solve a quadratic.
Scientific Calculator: general maths.
Matrix Calculator: matrix operations.

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