Factorising

Introduction to Factorising

Factorising is the process of breaking down an algebraic expression into its simplest factors.
It is the inverse operation of expanding brackets, reversing the process to return to a bracketed form.
At its most basic level, factorising involves identifying a common factor within an expression that can be taken outside of a bracket.
For instance, in the expression 4x + 8, the common factor is 4. Factorising this expression would give: 4(x+2).

When you have a single term, look for common number factors. The number factors of 12x for example are 1, 2, 3, 4, 6, and 12.
When considering what factors to use, think strategically about the rest of your equation. Certain factorisations may simplify subsequent steps.

Quadratic expressions are usually written in the form ax² + bx + c and can often be factorised.
To factorise a simple quadratic where a=1, find two numbers that multiply to give c (the constant term) and add to give b (the coefficient of x).
For example, to factorise x² + 5x + 6, find two numbers that multiply to 6 and add to 5. The numbers are 2 and 3, so the factorised form is (x + 2)(x + 3).

In algebra, the highest common factor (HCF) of two terms is the highest number that divides exactly into both numbers. This is hugely important when factorising.
To find the HCF, list all the factors of each number and select the highest number that appears on both lists.
Factorising with the HCF helps minimise expressions, making further operations more manageable.

Cubic expressions are a step up in difficulty but follow the same principles.
To factorise cubic expressions, look for common factors first, then attempt to factorise the resulting quadratic factor.
Take the expression: x³ - 3x² - x + 3: a common factor of x can be taken out giving x(x² - 3x -1); the quadratic factor can be factorised further resulting in the final factorised form as x(x - 1)(x + 3).

When factorising expressions that include negative terms, consider the negative sign as part of the term.
For example, in the expression 3x - 6, the common factor is 3, so the factorised form is 3(x - 2).

Be alert for common factors in the terms of your expression.
Check your answer by expanding the brackets in your factorised form. This will yield the original expression if you have factorised correctly.
Don’t be hurrying while solving complex expressions, take your time, stay calm and use systematic approach.
With regular practice, you will get better at spotting opportunities to simplify expressions via factorising.

Mastering factorising is a key skill in algebra which can greatly simplify problem-solving techniques.