IMO Practice Test — Factorisation
6 Questions • 15 min • Olympiad level
15:00
Question 1 of 6
hard
Factorise: x⁴ - 1
(x²+1)(x+1)(x-1)
(x²-1)(x²+1)
(x-1)⁴
(x+1)⁴
Explanation: x⁴-1=(x²-1)(x²+1)=(x-1)(x+1)(x²+1)
Question 2 of 6
hard
Factorise: 16x⁴ - 81y⁴
(4x²-9y²)(4x²+9y²)
(2x-3y)(2x+3y)(4x²+9y²)
Both A and B
Neither
Explanation: 16x⁴-81y⁴ = (4x²)²-(9y²)² = (4x²-9y²)(4x²+9y²) = (2x-3y)(2x+3y)(4x²+9y²)
Question 3 of 6
hard
Factorise: x³ + 1/x³
(x+1/x)(x²+1+1/x²)
(x+1/x)(x²-1+1/x²)
(x-1/x)(x²+1+1/x²)
(x-1/x)(x²-1+1/x²)
Explanation: Sum of cubes: a³+b³=(a+b)(a²-ab+b²) with a=x, b=1/x
Question 4 of 6
hard
If x + y = 5 and xy = 6, find x³ + y³
35
40
45
50
Explanation: x³+y³=(x+y)³-3xy(x+y)=125-3×6×5=125-90=35
Question 5 of 6
hard
Factorise: a² + b² - c² + 2ab
(a+b-c)(a+b+c)
(a-b+c)(a+b-c)
(a+b-c)²
(a-b+c)²
Explanation: a²+2ab+b²-c² = (a+b)²-c² = (a+b-c)(a+b+c)
Question 6 of 6
hard
The expression x⁴ + 4 can be factorised as:
(x²+2x+2)(x²-2x+2)
(x²+2)(x²-2)
(x²+2)²
Cannot factor
Explanation: Add and subtract 4x²: x⁴+4x²+4-4x² = (x²+2)²-(2x)² = (x²+2x+2)(x²-2x+2)