Polynomial Factoring Calculator

Enter a polynomial in one variable and this solver searches for a product of factors with whole-number or fraction coefficients. It takes out the greatest common factor, searches for rational roots, and can also find higher-degree factors, such as x⁴ + 4 = (x² + 2x + 2)(x² - 2x + 2). Each factor is listed with its multiplicity and roots.

How to use this tool

1. Type the polynomial, for example 6x^2 + x - 2 or 2x^3 - 8x. Products such as (x+1)(x^2-4) are expanded first.
2. Check the "Expanded polynomial" line to confirm the input was read the way you meant.
3. Read the factored form; a repeated factor is shown with a power.
4. The table gives the roots of each factor: exact fractions or surds when available, with numerical approximations otherwise.

Formula

P(x) = c · xᵏ · f₁(x)^m₁ · f₂(x)^m₂ · …

c is the constant factor (the GCF of the coefficients, with the sign), k is the power of x, and each fᵢ is a verified factor with integer coefficients and multiplicity mᵢ. Rational roots p/q obey the rational root theorem: p divides the constant term and q divides the leading coefficient. A quadratic factor whose discriminant b² − 4ac is not a perfect square has irrational or complex roots and stays as it is. Every factor is confirmed by exact integer division before it is shown.

Example

x³ - 6x² + 11x - 6: the possible rational roots are ±1, ±2, ±3 and ±6. P(1) = 1 − 6 + 11 − 6 = 0, P(2) = 8 − 24 + 22 − 6 = 0 and P(3) = 27 − 54 + 33 − 6 = 0, so the answer is (x - 1)(x - 2)(x - 3) with roots 1, 2 and 3.