Secondary 2 · Mathematics · Algebra
Expanding and factorising expressions
Use the distributive law to expand brackets such as 3(x + 4) and reverse the process by factorising out the highest common factor.
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These lessons were written around Australian Curriculum strands and matched to Singapore levels by age (Primary 1 is the first year of primary school). They are not aligned to MOE syllabuses or the PSLE, so check them against the syllabus your child is working towards.
Learning goals
Learners will
- Use the distributive law to expand brackets, including with negative numbers outside the brackets.
- Expand and simplify expressions with more than one set of brackets.
- Factorise expressions by taking out the highest common factor.
Success looks like
- I can expand 3(x + 4) to get 3x + 12.
- I can expand −2(a − 5) correctly to get −2a + 10.
- I can factorise 8a² − 12a fully as 4a(2a − 3).
The big idea
The distributive law says that a number outside brackets multiplies every term inside: a(b + c) = ab + ac. For example, 3(x + 4) = 3 × x + 3 × 4 = 3x + 12. Removing the brackets like this is called expanding.
An area model shows why. A rectangle 3 units wide and (x + 4) units long can be split into two parts: 3 by x (area 3x) and 3 by 4 (area 12). The total area is 3x + 12.
Take care with negative numbers outside the brackets. Multiply each term, using the sign rules (a negative times a negative is positive):
- −2(a − 5) = −2 × a + (−2) × (−5) = −2a + 10.
- −(x + 3) means −1(x + 3) = −x − 3.
To expand and simplify, expand each set of brackets, then collect like terms. 4(x + 3) + 2(x − 1) = 4x + 12 + 2x − 2 = 6x + 10.
Factorising reverses expanding. Find the highest common factor (HCF) of all the terms, write it outside the brackets, and write what is left inside. For 6x + 15, the HCF of 6x and 15 is 3, so 6x + 15 = 3(2x + 5).
The HCF can include pronumerals. In 8a² − 12a, both terms share 4 and a, so the HCF is 4a: 8a² − 12a = 4a(2a − 3). Always check by expanding your answer — you should get back to where you started. “Factorise fully” means take out the *highest* common factor, not just any factor.
Worked examples
Expanding with a negative
Expand −2(a − 5).
- Multiply −2 by a: −2a.
- Multiply −2 by −5: a negative times a negative is positive, so +10.
Answer: −2a + 10
Expanding and simplifying
Expand and simplify 4(x + 3) + 2(x − 1).
- 4(x + 3) = 4x + 12.
- 2(x − 1) = 2x − 2.
- Add them: 4x + 12 + 2x − 2.
- Collect like terms: 4x + 2x = 6x and 12 − 2 = 10.
Answer: 6x + 10
Factorising fully
Factorise 8a² − 12a fully.
- Numbers: the HCF of 8 and 12 is 4.
- Pronumerals: a² = a × a and a share one a.
- So the HCF is 4a. Divide each term by 4a: 8a² ÷ 4a = 2a and 12a ÷ 4a = 3.
- Write 4a(2a − 3). Check: 4a × 2a = 8a² and 4a × 3 = 12a. ✓
Answer: 4a(2a − 3)
Practice check
Have a go, then check your answers. Each answer comes with an explanation.
Answer guide for parents
Expand 5(x + 2).
5x + 10 — Multiply both terms by 5: 5 × x = 5x and 5 × 2 = 10.
Expand −3(y − 4).
−3y + 12 — −3 × y = −3y and −3 × −4 = +12.
What is the highest common factor of 12x and 18?
6 — The HCF of 12 and 18 is 6. The term 18 has no x, so x is not a common factor.
Factorise 10m − 15 fully.
5(2m − 3) — The HCF is 5: 10m ÷ 5 = 2m and 15 ÷ 5 = 3, giving 5(2m − 3). Check: 5 × 2m − 5 × 3 = 10m − 15.
Expand 4(2a + 3).
8a + 12 — 4 × 2a = 8a and 4 × 3 = 12.
Expand and simplify 2(x + 5) + 3(x − 1).
5x + 7 — 2x + 10 + 3x − 3. Collect like terms: 5x + 7.
Factorise 7x + 21.
7(x + 3) — The HCF of 7x and 21 is 7. 7x ÷ 7 = x and 21 ÷ 7 = 3.
Factorise 6p² + 9p fully.
3p(2p + 3) — The HCF of 6p² and 9p is 3p. 6p² ÷ 3p = 2p and 9p ÷ 3p = 3. Check: 3p × 2p + 3p × 3 = 6p² + 9p.
Watch out for
- Multiplying only the first term: writing 3(x + 4) = 3x + 4. Draw arrows from the outside number to every term inside.
- Sign errors with a negative outside, such as −2(a − 5) = −2a − 10. Remind the learner that −2 × −5 = +10.
- Not factorising fully, for example 8a² − 12a = 2(4a² − 6a). It is factorised, but not fully — the terms inside still share 2a. Ask “Is there anything else common inside the brackets?”
Tips for parents
- Ask the learner to check every factorisation by expanding it again. If it matches the original, it is right.
- Substitute a number to test an expansion: if x = 2, then 3(x + 4) = 18 and 3x + 12 = 18 — they match.
- Draw area-model rectangles together. Seeing the pieces makes the distributive law concrete.
Go further
- Use the distributive law for mental arithmetic: 7 × 98 = 7(100 − 2) = 700 − 14 = 686.
- Explore expanding two brackets, (x + 2)(x + 3), with an area model split into four rectangles.
- Create a matching card game with expanded expressions on some cards and factorised forms on others.
About 45 minutes
You'll need
- Paper and pencil
- Grid paper for drawing area models
- Coloured pencils
Key words
- Expand
- To remove brackets by multiplying each term inside by the term outside.
- Distributive law
- The rule a(b + c) = ab + ac: the outside term multiplies every term inside the brackets.
- Factorise
- To write an expression as a product, by taking a common factor outside brackets. It is the reverse of expanding.
- Highest common factor (HCF)
- The largest number or term that divides exactly into every term of an expression.
- Like terms
- Terms with exactly the same pronumeral part, such as 4x and −2x, which can be added or subtracted.
Still tricky?
For parents
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