Mathematics • Year 7 • Unit 2 • Lesson 10
Factorising — Common Factor
Build the basics: find the highest common factor (HCF) of the numbers and the variables, divide each term by the HCF, and write the answer as HCF × (remaining terms). Always check by expanding back.
1. I do — fully worked example
Read every line. Each step has a short reason on the right so you can see why, not just what.
Problem. Fully factorise 12x² + 8x.
Step 1 — Find the HCF of the numbers.
Factors of 12: {1, 2, 3, 4, 6, 12} Factors of 8: {1, 2, 4, 8} → HCF(12, 8) = 4
Reason: list all factors of each number, find the biggest one they share.
Step 2 — Find the HCF of the variable parts.
x² has power 2; x has power 1. Take the LOWEST power: x¹ = x.
Reason: x must appear in BOTH terms to be a common factor. Lowest power that's in both is x¹.
Step 3 — Overall HCF = 4 × x = 4x. Divide each term by 4x.
12x² ÷ 4x = 3x 8x ÷ 4x = 2
Reason: divide the coefficient by 4 and subtract powers of x for each term.
Step 4 — Write as HCF × (remaining terms in brackets), then check.
4x(3x + 2)
Check by expanding: 4x × 3x = 12x², and 4x × 2 = 8x. Sum: 12x² + 8x ✓ matches the original.
Answer: 12x² + 8x = 4x(3x + 2).
2. We do — fill in the missing steps
Same structure as Section 1, but with the working faded. Fill in each blank line. 4 marks
Problem. Fully factorise 6a² + 9a.
Step 1 — HCF of the numbers:
Factors of 6: {____, ____, ____, ____} Factors of 9: {____, ____, ____} → HCF = ______
Step 2 — HCF of the variable parts:
a² has power ____; a has power ____. Lowest power: a____ = ______
Step 3 — Overall HCF = ______. Divide each term:
6a² ÷ ______ = ______ 9a ÷ ______ = ______
Step 4 — Write as HCF × (remaining terms):
Final answer = ______________
3. You do — independent practice
Show your working under each question. The first four are foundation, the middle two are standard, and the last two are extension.
Foundation — single step
3.1 Factorise 4x + 8. 1 mark
3.2 Factorise 5x + 10. 1 mark
3.3 Factorise 6x + 9. 1 mark
3.4 Find the HCF of 8x and 12. 1 mark
Standard — combine two ideas
3.5 Factorise x² + 7x. (The HCF includes the variable.) 2 marks
3.6 Factorise 6a² + 9a. (Both a number and a variable in the HCF.) 2 marks
Extension — push your thinking
3.7 Fully factorise 12x² + 8x and verify your answer by expanding back. 3 marks
3.8 Fully factorise 5a²b + 10ab². (Two variables: each must appear in BOTH terms to be in the HCF.) 3 marks
How did this worksheet feel?
What I'll revisit before next class:
Section 2 — We do (6a² + 9a)
Step 1: Factors of 6: {1, 2, 3, 6}. Factors of 9: {1, 3, 9}. HCF = 3.
Step 2: a² has power 2; a has power 1. Lowest power: a1 = a.
Step 3: Overall HCF = 3a. 6a² ÷ 3a = 2a. 9a ÷ 3a = 3.
Step 4: Final answer = 3a(2a + 3). Check: 3a × 2a = 6a², 3a × 3 = 9a ✓.
3.1 — 4x + 8
HCF(4, 8) = 4. 4x ÷ 4 = x; 8 ÷ 4 = 2. Answer: 4(x + 2).
3.2 — 5x + 10
HCF(5, 10) = 5. 5x ÷ 5 = x; 10 ÷ 5 = 2. Answer: 5(x + 2).
3.3 — 6x + 9
HCF(6, 9) = 3. 6x ÷ 3 = 2x; 9 ÷ 3 = 3. Answer: 3(2x + 3).
3.4 — HCF of 8x and 12
Numbers: HCF(8, 12) = 4. Variables: x only appears in the first term (8x), so x cannot be in the HCF. Answer: HCF = 4.
3.5 — x² + 7x
Numbers: HCF(1, 7) = 1. Variables: x in both, lowest power x¹. HCF = x. Divide: x² ÷ x = x; 7x ÷ x = 7. Answer: x(x + 7).
3.6 — 6a² + 9a
HCF: numbers 3, variable a (lowest power). Overall HCF = 3a. Divide: 6a² ÷ 3a = 2a; 9a ÷ 3a = 3. Answer: 3a(2a + 3).
3.7 — 12x² + 8x
HCF: numbers 4, variable x (lowest power). Overall HCF = 4x. Divide: 12x² ÷ 4x = 3x; 8x ÷ 4x = 2. Answer: 4x(3x + 2).
Check by expanding: 4x × 3x = 12x², 4x × 2 = 8x. Sum: 12x² + 8x ✓.
3.8 — 5a²b + 10ab²
Numbers: HCF(5, 10) = 5. Variable a: in both (a² and a) → lowest power a. Variable b: in both (b and b²) → lowest power b. Overall HCF = 5ab. Divide: 5a²b ÷ 5ab = a; 10ab² ÷ 5ab = 2b. Answer: 5ab(a + 2b). Check: 5ab × a = 5a²b ✓ and 5ab × 2b = 10ab² ✓.