Rational Root Test Calculator

This rational root test calculator lists every possible rational root of a polynomial with integer coefficients, using the rational root theorem, turning root-finding into a short, manageable search instead of guesswork. You enter two numbers: the leading coefficient, the number in front of the highest power of x, and the constant term, the number with no x attached. The calculator finds every factor of the constant term and every factor of the leading coefficient, then builds each candidate by dividing a constant factor by a leading factor, with a plus and minus version of each. It returns the full list of candidates, the total number of candidates found, and a note showing how the constant and leading factors were combined to build that list. The theorem guarantees that any rational root of the polynomial appears somewhere on this list, though it does not tell you which candidates, if any, are genuine roots; you still need to substitute each one into the polynomial and check for a result of zero. This makes it a fast first step before factoring or solving a polynomial by hand, useful for algebra students and anyone tackling homework or exam revision. Try the default values of 1 and -6 to see the candidates plus or minus 1, 2, 3 and 6, then test each in your polynomial to find the true roots.

Leading coefficient

Constant term

Candidates: -6, -3, -2, -1, 1, 2, 3, 6

Number of candidates8

Constant and leading|const| factors x |lead| factors

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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