Basic factorising

Description

e.g. $4x+12=4(x+3)$

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.

Links

Expanding single brackets Prime factor decomposition