How this instrument works
The Rational Root Theorem says that any rational root of a MONIC polynomial (one whose leading coefficient is 1) with integer coefficients must itself be an integer that evenly divides the constant term. That's a filter, not a guarantee — it narrows down which integers are even worth testing as roots, without confirming that any particular candidate actually IS one.
This page applies that filter directly: given a polynomial's constant term and a candidate integer, it checks whether the candidate divides the constant term evenly. A remainder of 0 means the candidate stays on the shortlist worth testing further (by substituting it into the full polynomial); any nonzero remainder rules it out immediately, with no need to ever substitute it in at all.
This is a genuine time-saver for factoring polynomials by hand — instead of testing every integer near zero as a possible root, only the constant term's actual divisors need checking, which is often a much shorter list.
- Enter the polynomial's constant term into the d field.
- Enter the candidate integer you want to test into the r field.
- Read Test value: 0 means r divides d evenly (r stays a plausible candidate); nonzero rules r out.
- Repeat for every divisor of d to build the complete shortlist of candidates worth testing in the full polynomial.
Worked example — testing candidates against a constant of 6
For a polynomial ending in ...+6, testing whether 3 is a plausible root candidate means checking whether 3 divides 6 evenly: 6 mod 3=0, so 3 stays on the shortlist — though the theorem alone doesn't confirm 3 IS a root, only that it isn't immediately ruled out.
Testing whether 4 divides 6 evenly: 6 mod 4=2, nonzero, so 4 is ruled out immediately, with no need to substitute it into the polynomial at all. Testing whether 4 divides −9 evenly: −9 mod 4=3, also nonzero, so 4 is ruled out as a candidate for a polynomial ending in ...−9 as well.
Questions
What is the Rational Root Theorem?
A rule stating that any rational root of a monic polynomial with integer coefficients must itself be an integer dividing the constant term evenly — a filter for narrowing down candidates, not a method for finding roots directly.
Does passing this test mean a number is definitely a root?
No — it only means the candidate is WORTH testing further by substituting it into the full polynomial. Many numbers that divide the constant term evenly still turn out not to be actual roots once tested.
Why does this only apply to monic polynomials?
The simplified version used here assumes a leading coefficient of 1; the theorem's general form for non-monic polynomials also involves the leading coefficient's own divisors, a more involved case this page doesn't cover.
What should I do once a candidate passes this test?
Substitute it into the full polynomial and check whether the result is actually zero — passing this divisibility test only means it's plausible, not confirmed, so the final check still has to be done separately.
Can a negative number be a candidate root?
Yes — negative integers that divide the constant term evenly are just as valid to test as positive ones; the theorem doesn't restrict candidates to positive divisors alone.