Factorising is the reverse of expanding brackets — you take out the highest common factor from every term. For example, $4x+12=4(x+3)$.
We look at each term in the expression and find the highest common factor of each. This isn't necessarily just a number, it could be a letter or combination too.
$4x^{2}+ 12x=4x(x+3)$
Notice that in the above example, we could have pulled 2 out as a factor but this then wouldn't be fully factorised. We can tell because the terms in the brackets have common factors (2 and $x$) in them:
$4x^2+12x=2(2x^2+6x)$
The cherry tree method
A cherry tree helps you find the HCF systematically — especially useful when the terms include algebraic parts that are easy to miss.
Split each term into its factors like a factor tree. The factors that appear in every tree are the HCF (what goes outside the bracket). The leftover factors — the remaining "cherries" — go inside the bracket.
Example: Factorise $6x^2 + 9x$
Draw a cherry tree for each term:
- $6x^2 = 3 \times 2 \times x \times x$
- $9x = 3 \times 3 \times x$
Common to both: $3$ and $x$, so the HCF is $3x$.
Remaining cherries:
- $6x^2 \div 3x = 2x$
- $9x \div 3x = 3$
$$6x^2 + 9x = 3x(2x + 3)$$
You can always check by expanding back out.