Work through the levels in order. Type your answer and press Check — open the worked solution if you get stuck. Answers like -6, 3/4 or 0.25 are typed exactly as shown. A ⭐ Bonus Challenge at the end is extra credit — skip it if the rest was hard work.
Level 1 · Warm-up
1 Factorise fully 24a2b + 16ab.
Show worked solution
HCF = 8ab. 24a2b ÷ 8ab = 3a and 16ab ÷ 8ab = 2: 8ab(3a + 2). 4ab(6a + 4) is an incomplete HCF (6a + 4 still shares 2); 8ab(3a + 2b) divides 16ab by 8ab wrongly; 8a(3ab + 2b) is an incomplete HCF (b is in both terms).
2 Factorise −12y + 30 by taking out −6.
Show worked solution
−12y ÷ (−6) = 2y and 30 ÷ (−6) = −5, so −12y + 30 = −6(2y + (−5)) = −6(2y − 5). Check: −6 × −5 = +30 ✓.
3 Factorise ab + 3a + 4b + 12 by grouping, then use your factors to work out its value when a = 6 and b = 17.
Show worked solution
a(b + 3) + 4(b + 3) = (b + 3)(a + 4). With a = 6 and b = 17: 20 × 10 = 200. Check: 102 + 18 + 68 + 12 = 200 ✓.
Level 2 · Core
4 Factorise 3x − 3y − ax + ay.
Show worked solution
3(x − y) − a(x − y) = (x − y)(3 − a). (x − y)(3 + a) is the sign trap (it expands to +ax); (x + y)(3 − a) gets the bracket wrong; 3(x − y) − a(x − y) stops before taking out the matching bracket.
5 Factorise −15a2b + 10ab2 − 5ab by taking out −5ab.
Show worked solution
−15a2b ÷ (−5ab) = 3a, 10ab2 ÷ (−5ab) = −2b, −5ab ÷ (−5ab) = 1. So −5ab(3a + (−2)b + 1) = −5ab(3a − 2b + 1).
6 −18m4n + 27m2n3 is factorised fully by taking out a negative common factor, and the bracket left is (2m2 − 3n2). What number is in front of the common factor?
Show worked solution
HCF of 18 and 27 is 9; lowest powers are m2 and n. Taking out −9m2n: −18m4n ÷ (−9m2n) = 2m2 and 27m2n3 ÷ (−9m2n) = −3n2. So the factor is −9m2n and the number is −9.
Level 3 · Challenge
7 Three statements:
1 ab − a − b + 1 = (a − 1)(b − 1)
2 2x2 − 6xy + 5x − 15y = (x − 3y)(2x + 5)
3 mn − 4m + 8 − 2n = (m − 2)(n + 4)
Which statements are correct?
1 ab − a − b + 1 = (a − 1)(b − 1)
2 2x2 − 6xy + 5x − 15y = (x − 3y)(2x + 5)
3 mn − 4m + 8 − 2n = (m − 2)(n + 4)
Which statements are correct?
Show worked solution
a(b − 1) − 1(b − 1) = (a − 1)(b − 1) (1 ✓). 2x(x − 3y) + 5(x − 3y) = (x − 3y)(2x + 5) (2 ✓). mn − 4m − 2n + 8 = m(n − 4) − 2(n − 4) = (n − 4)(m − 2), but (m − 2)(n + 4) expands to mn + 4m − 2n − 8 (3 ✗). 1 and 2.
8 Factorise x3 + 4x2 − 3x − 12 by grouping.
Show worked solution
x2(x + 4) − 3(x + 4) = (x + 4)(x2 − 3).
9 Use grouping to work out 17 × 23 + 17 × 77 − 13 × 23 − 13 × 77 without a calculator.
Show worked solution
17(23 + 77) − 13(23 + 77) = (23 + 77)(17 − 13) = 100 × 4 = 400.