Factorisation definition
Factorisation, also called factorising, is the process of rewriting a number or algebraic expression as a product of factors. In algebra, this usually means putting an expression into brackets. For example, 3x + 6 can be factorised as 3(x + 2). The factorised form means 3 multiplied by x + 2. If you expand the brackets, you return to the original expression.
Factorisation is the reverse of expanding brackets. Expanding changes 4(y + 5) into 4y + 20. Factorising changes 4y + 20 back into 4(y + 5). This reverse relationship is useful because students can check a factorisation by expanding it again. If the expanded result matches the original expression, the factorisation is correct.
Factorisation is used in algebra, expressions, equations, quadratics, functions, graphs, simplifying algebraic fractions, proof and calculus. At GCSE level, students learn to factorise by taking out a common factor, factorising quadratics and recognising special forms such as the difference of two squares. At A-Level, factorisation supports polynomial division, solving equations, integration, partial fractions and curve sketching.
The simplest type of factorisation uses a common factor. In the expression 5x + 10, both terms are divisible by 5. Therefore, 5x + 10 factorises to 5(x + 2). In the expression 6a + 9, the highest common factor is 3, so it factorises to 3(2a + 3). Choosing the highest common factor gives the most complete factorisation.
Quadratic factorisation is another important skill. For example, x squared + 5x + 6 can be factorised as (x + 2)(x + 3). This works because 2 times 3 equals 6 and 2 plus 3 equals 5. When the brackets are expanded, the result is x squared + 3x + 2x + 6, which simplifies to x squared + 5x + 6. This shows why multiplication and addition both matter when factorising quadratics.
Factorisation can help solve equations. If x squared + 5x + 6 = 0, factorising gives (x + 2)(x + 3) = 0. This means x + 2 = 0 or x + 3 = 0, so x = -2 or x = -3. This method works because if two factors multiply to make zero, at least one of the factors must be zero. This is called the zero product rule.
A special factorisation is the difference of two squares. For example, x squared - 9 factorises to (x - 3)(x + 3) because 9 is 3 squared. In general, a squared minus b squared equals (a - b)(a + b). Recognising this pattern saves time and helps students solve harder algebraic questions efficiently.
Related ideas include factors, common factors, expanding brackets, simplifying, quadratic equations, algebraic fractions and identities. A strong definition of factorisation helps students understand that factorising is not just adding brackets randomly. It is rewriting an expression as multiplication of factors while keeping the value the same. In exams, always check your answer by expanding the brackets again.