Factoring Calculator

Factor any polynomial — quadratics, trinomials, differences of squares, and higher-degree expressions — with the greatest common factor pulled out and every step shown.

Factored form —

How to Factor a Polynomial

Enter a polynomial in expanded form and the calculator breaks it into factors, showing the reasoning.

1

Pull out the greatest common factor

First the calculator divides out anything common to every term — a shared number and the lowest power of the variable — so 6x² + 12x becomes 6x(x + 2) before anything else.

2

Factor what remains

It recognises special forms like the difference of squares a² − b² = (a − b)(a + b) and perfect-square trinomials, and factors quadratics into two binomials.

3

Handle higher degrees with rational roots

For cubics and beyond it finds a rational root, divides out that linear factor, and repeats until only irreducible pieces remain. Want the roots directly instead? The Quadratic Formula Calculator solves ax² + bx + c = 0, and the Polynomial Calculator multiplies factors back out to check.

Frequently Asked Questions

How do you factor a polynomial?
Start by writing the polynomial in expanded standard form and pulling out the greatest common factor of every term — both the numeric GCF and the lowest power of the variable. Then factor what remains: a quadratic ax² + bx + c splits into two binomials, a difference of squares a² − b² becomes (a − b)(a + b), and for higher degrees you find a rational root and divide it out, repeating until only irreducible pieces are left. This calculator does all of that automatically and lists each step.
What can this factoring calculator factor?
It factors single-variable polynomials over the rational numbers: it pulls out the greatest common factor, factors quadratics and trinomials (including non-monic ones like 6x² + 11x − 10), recognises perfect-square trinomials and differences of squares, and uses the rational-root theorem plus synthetic division to factor cubics, quartics, and higher. Enter the polynomial in expanded form, such as x^2 + 5x + 6 or 2x^3 - 4x^2 - 22x + 24.
What does it mean when a polynomial can't be factored?
A polynomial that has no factors with rational coefficients other than itself and 1 is called irreducible (or prime) over the rationals. For example, x² + 1 and x² − 2 cannot be broken into simpler factors using whole-number or fractional coefficients. When that happens the calculator reports the polynomial as already irreducible rather than forcing an incorrect factorisation.
How do you factor a trinomial like ax² + bx + c?
For a trinomial ax² + bx + c you look for two binomials (px + q)(rx + s) whose product returns the original. When a = 1 you need two numbers that multiply to c and add to b; when a is not 1 you factor by grouping or test the rational roots. This calculator finds the exact factors when they exist over the rationals and shows the roots it used, so you can follow the reasoning as well as read the answer.

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