To factorise with a common factor, take the highest factor shared by every term outside a bracket, and leave what remains inside. It is expanding in reverse: if you expand the answer, you must get the original. This is usually the first thing to check in any factorising question.
It follows on from expanding brackets with negative terms and prepares you for factorising quadratics.
What counts as the highest common factor?
For terms like 12x²y and 18xy², work on the number and on each letter separately.
The number part: the highest common factor of 12 and 18 is 6. The letter x: the lowest power in both terms is x¹. The letter y: the lowest power is y¹.
Put them together and the factor is 6xy.
“Highest” means you take out everything that is shared, so nothing shared remains in the bracket.
How to factorise, step by step
- Find the HCF of the numbers.
- Find each letter that appears in every term, and choose its lowest power.
- Write the common factor outside the bracket.
- Divide each term by the factor and write the results inside the bracket, keeping signs.
- Expand to check, and confirm no further common factor is left inside.
Worked example
Factorise completely 12x²y − 18xy²
Step 1, numbers: the HCF of 12 and 18 is 6.
Step 2, letters: both terms contain x and y. The lowest power of x is 1 and the lowest power of y is 1. The common factor is 6xy.
Step 3, divide each term:
- 12x²y ÷ 6xy = 2x
- −18xy² ÷ 6xy = −3y
Step 4, answer:
6xy(2x − 3y)
Check: expand back. 6xy × 2x = 12x²y and 6xy × (−3y) = −18xy². Both terms match the original. Inside the bracket, 2x and 3y share nothing, so the factorisation is complete.
The mistake to watch for
A common slip is to forget the 1 that remains when a whole term is taken out.
Mistaken answer: 8x² + 8x = 8x(x)
The student took out 8x from both terms but wrote only x inside. Expanding gives 8x², not 8x² + 8x, so a term has disappeared.
The correction is to divide every term, including the one that equals the factor. 8x² ÷ 8x = x, and 8x ÷ 8x = 1. So the answer is 8x(x + 1).
A second slip is partial factorising, such as 12x² − 18x = 3x(4x − 6). Expanding does give the original, but the bracket still has a common factor of 2, so the answer is not complete. The full factorisation is 6x(2x − 3).
Check yourself
Try these without a calculator, then open each answer.
1. Factorise completely 15ab − 10b²
Show answer
Numbers: HCF of 15 and 10 is 5. Letter b appears in both terms with lowest power 1. The common factor is 5b.
15ab ÷ 5b = 3a, and −10b² ÷ 5b = −2b.
5b(3a − 2b)
2. Factorise completely 4x³ + 12x²
Show answer
HCF of 4 and 12 is 4. Lowest power of x is 2, so x². The factor is 4x².
4x³ ÷ 4x² = x, and 12x² ÷ 4x² = 3.
4x²(x + 3)
Check: 4x² × x = 4x³ and 4x² × 3 = 12x².
3. Factorise −9p − 6, taking out a negative factor
Show answer
The HCF of 9 and 6 is 3. Taking out −3 changes the signs inside: −9p ÷ (−3) = 3p, and −6 ÷ (−3) = +2.
−3(3p + 2)
Check: −3 × 3p = −9p and −3 × 2 = −6.
Where this leads next
With common factors secure, move on to factorising a quadratic with integer factors, where the common factor is the first thing to check. The non-calculator working trainer can help with finding the highest common factor of awkward numbers.
If you can follow each worked example but miss the full common factor on a fresh question, a teacher can build a step-by-step routine with you in online one-to-one Mathematics tuition.