How to Factorise Algebraic Expressions
Simple factorising is a fundamental GCSE and IGCSE Maths skill where expressions are rewritten as a product of factors by taking out the greatest common factor. This topic is the reverse of expanding and appears throughout algebra, equations, and fractions. In this guide you will learn how to identify the greatest common number, choose the lowest shared power of each letter, and write clear, correct factorised forms using worked examples.
What do we factor out?
Find the shared number, then the shared letters (using the lowest powers). Example:
Tip: if the first term is negative, factoring a negative makes the bracket tidier.
Core ideas to remember
1) Greatest common number
Divide coefficients by the largest number that fits all.
2) Lowest shared power
For \(x^3\) and \(x^2\) the shared power is \(x^2\).
3) Consider a negative GCF
Make the first term in the bracket positive.
Worked examples
Example A — number only
\[ \begin{aligned} & 28a - 42 \\ =\;& 14(2a - 3) \end{aligned} \]Example B — number & one letter
\[ \begin{aligned} & 12x + 18x^2 \\ =\;& 6(2x + 3x^2) \\ =\;& 6x(2 + 3x) \end{aligned} \]Example C — two letters
\[ \begin{aligned} & 14x^2y - 21xy^2 \\ =\;& 7xy(2x - 3y) \end{aligned} \]Example D — negative first term
\[ \begin{aligned} & -9m^2 + 6m \\ =\;& -3m(3m - 2) \end{aligned} \]Example E — three terms
\[ \begin{aligned} & 28a^2b - 12ab^2 + 8ab \\ =\;& 4ab(7a - 3b + 2) \end{aligned} \]Example F — check by expanding
\[ \begin{aligned} & 3x^2y + 9xy^2 \\ =\;& 3xy(x + 3y) \end{aligned} \]Common mistakes
- Not taking the greatest common number.
- Using the highest instead of the lowest common power of each letter.
- Missing a negative factor when it tidies the bracket.
- Dropping a letter/sign when writing the bracket.
- Not checking by re-expanding.
🔑 Key points
- GCF = greatest common number × lowest power of each common letter.
- Consider factoring out a negative for a neat bracket.
- Write the outside factor once; put the simplified remainder inside brackets.
- Always verify by expanding back.