About the Factoring Polynomials Calculator
This factoring polynomials calculator factors a polynomial in one variable completely over the integers and shows how. It first pulls out the greatest common factor (including any power of x), then recognises the standard patterns taught in algebra — difference of squares, perfect square trinomials, sum and difference of cubes, factoring by grouping and the ac method for trinomials — and finishes with the rational root theorem for anything of higher degree.
Type the polynomial the way you would write it, for example 2x^3 - 8x, 6x^2 + 11x - 10 or x^4 - 5x^2 + 4. Expressions with brackets are expanded first, so you can also check that a factorisation you did by hand multiplies back to the original. The result lists every factor, the rational roots with their multiplicity, and any factor that cannot be broken down further over the integers.
It is built for algebra and precalculus students and for teachers making answer keys. "Over the integers" means factors keep whole-number coefficients: x² − 2 is left as it is, even though it equals (x − √2)(x + √2) over the real numbers.
How to use the factoring polynomials calculator
- 1Type the polynomial using ^ for powers, e.g. 3x^3 - 12x.
- 2Read the factored form at the top.
- 3Check which method was used and follow the steps table.
- 4Use the rational roots to solve the equation polynomial = 0.
Formula and method
Factoring always starts with the greatest common factor of all the terms, including the lowest power of the variable. What remains is checked against the special patterns: a difference of two squares, a perfect square trinomial, a sum or difference of cubes, four-term polynomials that factor by grouping, and trinomials ax² + bx + c, which factor when two integers multiply to a·c and add to b (the ac method).
For general polynomials the rational root theorem lists every possible rational root p/q, where p divides the constant term and q divides the leading coefficient. Each candidate that makes the polynomial zero gives a factor (qx − p), which is divided out by synthetic division; repeating finds all rational roots and their multiplicities. Any leftover of degree four or more is tested for integer quadratic factors, and whatever remains is irreducible over the integers.
- GCF
- Greatest common factor of all terms
- p/q
- Candidate rational root (p | constant, q | leading coefficient)
- a·c
- Product used to split the middle term of a trinomial
Worked examples
Trinomial with a leading coefficient
a·c = 6·(−10) = −60. The numbers 15 and −4 multiply to −60 and add to 11, so 6x² + 15x − 4x − 10 = 3x(2x + 5) − 2(2x + 5) = (2x + 5)(3x − 2).
GCF then difference of squares
The GCF is 2x, leaving x² − 4, a difference of squares: (x − 2)(x + 2). So 2x³ − 8x = 2x(x + 2)(x − 2).
Difference of cubes
x³ − 27 = x³ − 3³ = (x − 3)(x² + 3x + 9). The quadratic factor has discriminant 9 − 36 < 0, so it does not factor further over the integers.
Quadratic in x²
With u = x², this is u² − 5u + 4 = (u − 1)(u − 4) = (x² − 1)(x² − 4), and each difference of squares splits again into (x − 1)(x + 1)(x − 2)(x + 2).
Factoring by grouping
Group as x²(x + 3) − 4(x + 3) = (x + 3)(x² − 4), and x² − 4 = (x − 2)(x + 2).
Frequently asked questions
What is the first step in factoring a polynomial?+
Always factor out the greatest common factor first — the largest number and the lowest power of the variable that divide every term. For 4x³ + 8x² that is 4x², leaving 4x²(x + 2).
How do you factor a trinomial when a is not 1?+
Use the ac method: multiply a and c, find two numbers that multiply to ac and add to b, split the middle term with them, then factor by grouping. For 6x² + 11x − 10, ac = −60 and the numbers are 15 and −4.
How do you factor a cubic polynomial?+
Look for a GCF, grouping, or a sum or difference of cubes. Otherwise use the rational root theorem to find one root r, divide by (x − r) with synthetic division, and factor the remaining quadratic.
What does irreducible over the integers mean?+
It means the polynomial cannot be written as a product of lower-degree polynomials with integer coefficients. x² + 1 and x² − 2 are irreducible over the integers, though x² − 2 factors over the reals.
Can a sum of squares be factored?+
A sum of two squares like x² + 9 cannot be factored over the real numbers. Some higher sums can, though: x⁴ + 4 = (x² − 2x + 2)(x² + 2x + 2), which this calculator finds.