Mathematics • Year 10 • Unit 2 • Lesson 4
Factorising Common Factors — Mixed Challenge
Pull together every idea from Lesson 4: HCF of numbers, HCF of letters (lowest powers), pulling out negatives, and factorising by grouping. Pick the right tool, spot another student's slip, then design an expression whose HCF is a specific target.
1. Mixed problems — choose the right tool
Show your working. 3 marks each
1.1 Find the HCF of 12x²y and 18xy².
1.2 Fully factorise 16x − 24.
1.3 Fully factorise 9x²y + 12xy + 6xy².
1.4 Fully factorise 7a²b − 14ab²+ 21ab.
1.5 Factorise by grouping: 2x + 6 + xy + 3y.
1.6 A rectangle has area 20a²b + 30ab². (a) Factorise. (b) If one side is 10ab, what is the other side? (c) Evaluate the area when a = 2 and b = 1.
2. Find the mistake
Another Year 10 student has tried to fully factorise 24x³y² − 36x²y³. Their working is shown below. Exactly one line contains a mistake. Spot it, explain why it's wrong, then re-do the working correctly. 3 marks
Student's working — fully factorise 24x³y² − 36x²y³:
Line 1: Number HCF: HCF(24, 36) = 12
Line 2: Letter HCF: lowest power of x is x²; lowest power of y is y³
Line 3: HCF = 12x²y³
Line 4: 24x³y² ÷ 12x²y³ = 2x ÷ y 36x²y³ ÷ 12x²y³ = 3
Line 5: = 12x²y³(2x/y − 3)
(a) Which line contains the mistake?
(b) Explain in one or two sentences why that line is wrong.
(c) Write out the corrected working in full, including the corrected fully factorised final answer.
Stuck? For the letter HCF, take the LOWEST power of each common letter — the lower of y² and y³ is y², not y³.3. Open-ended challenge — design an expression with a target HCF
This question has many valid answers. Be creative but show every number. 4 marks
3.1 Design a three-term algebraic expression in two variables (e.g. x and y) whose HCF is exactly 6xy — no smaller, no larger.
In your submission, include:
(i) Your three-term expression.
(ii) A short check that 6xy divides every term.
(iii) A short check that no factor larger than 6xy divides all three terms (e.g. the inner terms in the bracket must not share another common factor).
(iv) The fully factorised form.
Bonus: Include a negative coefficient somewhere in your expression.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — HCF of 12x²y and 18xy²
Number HCF = 6. Letter HCF: lowest power of x is x¹; lowest power of y is y¹. So HCF = 6xy.
1.2 — Fully factorise 16x − 24
HCF(16, 24) = 8. 16x ÷ 8 = 2x; 24 ÷ 8 = 3. Answer: 8(2x − 3).
1.3 — Fully factorise 9x²y + 12xy + 6xy²
Number HCF = 3. Letter HCF = xy. Total HCF = 3xy.
9x²y ÷ 3xy = 3x; 12xy ÷ 3xy = 4; 6xy² ÷ 3xy = 2y. Answer: 3xy(3x + 4 + 2y).
1.4 — Fully factorise 7a²b − 14ab² + 21ab
Number HCF = 7. Letter HCF = ab. Total HCF = 7ab.
7a²b ÷ 7ab = a; 14ab² ÷ 7ab = 2b; 21ab ÷ 7ab = 3. Answer: 7ab(a − 2b + 3).
1.5 — Factorise by grouping 2x + 6 + xy + 3y
Pair up: (2x + 6) + (xy + 3y).
Factor each pair: 2(x + 3) + y(x + 3).
Pull out (x + 3): (x + 3)(2 + y).
1.6 — Rectangle 20a²b + 30ab²
(a) Number HCF = 10, letter HCF = ab → 10ab(2a + 3b).
(b) Other side = (2a + 3b).
(c) When a = 2, b = 1: area = 10(2)(1)(2(2) + 3(1)) = 20 × (4 + 3) = 20 × 7 = 140 square units.
2 — Find the mistake
(a) The mistake is on Line 2 (and it propagates through Lines 3, 4, 5).
(b) The letter HCF takes the lowest power of each common letter. The lowest power of y between y² and y³ is y², not y³. The student picked the higher power.
(c) Corrected working:
Number HCF: 12.
Letter HCF: lowest powers — x² and y². So letter HCF = x²y².
Total HCF = 12x²y².
24x³y² ÷ 12x²y² = 2x; 36x²y³ ÷ 12x²y² = 3y.
Fully factorised: 12x²y²(2x − 3y).
Lesson § "Common Pitfalls" warns specifically about this lowest-power rule.
3 — Open-ended challenge (sample solutions)
Sample 1 — HCF exactly 6xy
Expression: 12x²y + 18xy² + 30xy.
(i) Each term checks: 12x²y ÷ 6xy = 2x ✓; 18xy² ÷ 6xy = 3y ✓; 30xy ÷ 6xy = 5 ✓.
(ii) Inside the bracket the three terms (2x, 3y, 5) share no common factor (GCD(2,3,5) = 1 and the letters do not repeat), so 6xy cannot be enlarged.
(iii) Fully factorised: 6xy(2x + 3y + 5).
Sample 2 — with a negative (bonus)
Expression: 12x²y − 18xy² + 6xy.
Each ÷ 6xy gives 2x, −3y, 1 → coprime ✓.
Fully factorised: 6xy(2x − 3y + 1).
Marking: 1 for a valid three-term expression, 1 for showing 6xy divides every term, 1 for arguing no larger factor divides all three (coprime inner terms), 1 for the fully factorised form. Full marks for any valid construction.