Factorisation Calculator
Factorise quadratic equations, polynomials, and algebraic expressions with step-by-step solutions. Supports common factors, difference of squares, grouping, and the AC method.
Algebra Factoriser
Factorise expressions completely with detailed steps
Enter the polynomial you want to factorise. Use ^ for powers, * for multiplication.
The main variable in your expression
Let the calculator detect the best method, or specify one
View the complete step-by-step factorisation process with explanations
Your Factored Expression
Step-by-step factorisation with final result
Enter your expression and click Factorise Expression to see the solution with detailed steps.
Common Factorisation Formulas
Essential algebraic identities and formulas for factorising polynomials.
| Formula Type | Standard Form | Factored Form |
|---|---|---|
| Common Factor | ab + ac | a(b + c) |
| Difference of Squares | a² – b² | (a – b)(a + b) |
| Perfect Square (Plus) | a² + 2ab + b² | (a + b)² |
| Perfect Square (Minus) | a² – 2ab + b² | (a – b)² |
| Sum of Cubes | a³ + b³ | (a + b)(a² – ab + b²) |
| Difference of Cubes | a³ – b³ | (a – b)(a² + ab + b²) |
| Quadratic (Roots) | ax² + bx + c | a(x – r₁)(x – r₂) |
| Grouping (4 terms) | ac + ad + bc + bd | (a + b)(c + d) |
Factorisation Calculator FAQ
Everything you need to know about factorising algebraic expressions and solving polynomials.
Factorisation is the process of breaking down a mathematical expression into a product of simpler expressions (factors) that, when multiplied together, give the original expression. It is essentially the reverse of expanding brackets. For example, the factorisation of x² + 5x + 6 is (x + 2)(x + 3).
To factorise a quadratic expression like ax² + bx + c, find two numbers that multiply to give ‘ac’ and add to give ‘b’. Rewrite the middle term using these numbers, then factorise by grouping. For simple quadratics where a=1, find two numbers that multiply to ‘c’ and add to ‘b’. For example, x² + 5x + 6 becomes (x + 2)(x + 3) because 2 × 3 = 6 and 2 + 3 = 5.
The difference of two squares is a special algebraic form: a² – b². It can always be factorised into (a – b)(a + b). For example, x² – 25 becomes (x – 5)(x + 5). Note that the sum of two squares (a² + b²) cannot be factorised over the real numbers.
The AC method is used for quadratics in the form ax² + bx + c where a ≠ 1. Multiply ‘a’ and ‘c’ together. Find two numbers that multiply to ‘ac’ and add to ‘b’. Split the middle term ‘bx’ into two terms using these numbers, then factorise the resulting four terms by grouping. For example, 2x² + 7x + 3: ac = 6, factors are 1 and 6. Rewrite as 2x² + 6x + x + 3, then group to get (2x + 1)(x + 3).
No, not all polynomials can be factorised into simpler polynomials with integer (or rational) coefficients. Polynomials that cannot be broken down further are called ‘prime’ or ‘irreducible’ polynomials. For example, x² + 1 is irreducible over the real numbers because there are no real numbers that multiply to 1 and add to 0.
Factorising by grouping is a technique used for polynomials with four or more terms. You group the terms into pairs, factor out the greatest common factor from each pair, and then factor out the common binomial bracket. For example, ac + ad + bc + bd becomes a(c + d) + b(c + d) = (a + b)(c + d).
To check if your factorisation is correct, simply expand (multiply out) your factors. If you get back the original expression exactly, your factorisation is correct. For example, if you factorised x² + 5x + 6 as (x + 2)(x + 3), expanding gives x² + 3x + 2x + 6 = x² + 5x + 6, which matches the original.
A common factor is a number, variable, or expression that divides evenly into every term of a polynomial. The first step in any factorisation should be to look for the Greatest Common Factor (GCF). For example, in 6x² + 9x, the GCF is 3x, so it factorises to 3x(2x + 3).