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 ✓.