Expanding multiplies every term inside a bracket by the factor outside. Factorising does the reverse: it pulls a common factor out and leaves a bracket behind.
How does expanding work?
Multiply the outside factor by each term inside, one at a time, keeping signs with their terms.
For example, 3(2x − 5) becomes 3 × 2x and 3 × (−5). That gives 6x − 15.
Test it with x = 4. The original is 3(8 − 5) = 9, and the answer is 24 − 15 = 9.
How does factorising work?
Look for the highest common factor of all terms. Write it outside a bracket, then divide each term by it to fill the bracket.
Take 12x + 18. The highest number dividing both 12 and 18 is 6. Dividing gives 2x and 3, so 12x + 18 = 6(2x + 3).
Check by expanding: 6 × 2x = 12x and 6 × 3 = 18. This works.
Worked example
Take the original expression 4a² − 10a and factorise it.
| Step | Working |
|---|---|
| Numbers | 4 and 10 share the factor 2 |
| Letters | a² and a share the letter a |
| Common factor | 2a |
| Fill bracket | 4a² ÷ 2a = 2a and −10a ÷ 2a = −5 |
| Answer | 2a(2a − 5) |
Expanding back: 2a × 2a = 4a² and 2a × (−5) = −10a. The check matches the start.
What does the common mistake look like?
A student expands 5(x + 3) as 5x + 3. Only the first term was multiplied. The correct expansion is 5x + 15.
A quick test catches it. Let x = 1. Then 5(1 + 3) = 20, but 5(1) + 3 = 8. Two different answers mean the expansion is wrong.
A similar slip in factorising is taking out a factor that is too small. For 12x + 18, writing 3(4x + 6) is not wrong, but it is incomplete: the bracket still has a common factor of 2. Always check that nothing more can come out.
Steps to follow
- To expand, multiply the outside factor by every term inside, signs included.
- To factorise, find the highest common factor of the numbers, then any shared letter.
- Always check by expanding your factorised answer or substituting a small number.
Self-check
Expand −2(3y − 4) and factorise 15p + 25.
Answer
Expanding: −2 × 3y = −6y and −2 × (−4) = +8, so −6y + 8. Check with y = 1: −2(3 − 4) = 2, and −6 + 8 = 2.
Factorising: the highest common factor of 15 and 25 is 5, so 15p + 25 = 5(3p + 5). Expanding back gives 15p + 25.
Where to go next
Brackets appear in almost every equation, so the next skill is solving a linear equation step by step. If simplifying after expanding causes trouble, revisit collecting like terms. The algebra readiness hub shows the whole sequence.