How this calculator works
For an integer polynomial, any reduced rational zero p/q has numerator dividing the constant coefficient and denominator dividing the leading coefficient. Remove zero roots first so a zero constant does not require an infinite divisor list. Every remaining candidate is checked by denominator-cleared Horner evaluation with exact integers. Actual nonzero roots are listed once, not with multiplicity.
Formula
If P(p/q) = 0 and gcd(p,q) = 1, p divides the constant and q divides the leading coefficient. Check q^n P(p/q) = 0 using integer arithmetic.
Worked example
For x^2 - 3x + 2, candidates are -2, -1, 1 and 2; direct substitution leaves roots 1 and 2. For 2x^2 - x = x(2x - 1), the roots are 0 and 1/2.
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Frequently asked questions
Does this find every real root?
No. Irrational and non-real roots are outside the rational-root theorem.
Why require integer coefficients?
The divisibility theorem applies directly to integer coefficients. Clear fractional denominators before entering the polynomial.
Does a candidate always work?
No. Divisibility supplies possibilities, then exact substitution determines which are actual roots.