About this tool
The Polynomial Factoring Calculator rewrites a single-variable polynomial as a product of simpler factors with integer or rational coefficients. It works in the standard order: first it pulls out the greatest common factor — the gcd of all coefficients together with the lowest common power of the variable — so 2x³ − 8x becomes 2x(x² − 4). Then it searches for rational roots using the Rational Root Theorem, testing every candidate ±p/q where p divides the constant term and q divides the leading coefficient.
Each rational root r = p/q found by synthetic division contributes a factor (qx − p), and the quotient is reduced and searched again. Patterns like the difference of squares a² − b² = (a − b)(a + b) fall out of this process automatically as two rational roots, so x² − 4 factors to (x − 2)(x + 2). Whatever remains after all rational roots are removed is a quadratic or higher factor with no rational roots; it is irreducible over the rationals and is left in place and clearly labeled.
All coefficients are handled as exact fractions and everything runs locally in your browser with an eval-free parser. Enter a polynomial such as '2x^3 - 8x' or 'x^2 - 5x + 6' and read the extracted GCF, each linear factor, any irreducible remainder, and the fully factored product form.