The function f(x) = ax³ + 4x² + bx – 2, where a and b are constants, is such that 2x – 1 is a factor. Given that the remainder when f(x) is divided by x - 2 is twice the remainder when f(x) is divided by x + 1, find the value of a and of b.

Mark scheme

Marking points from the official mark scheme (Oct/Nov 2013). The mark or mark type is shown in brackets.

1. [M1] for substitution of x = 1/2 into f(x)
2. [A1] for correct equation in any form (e.g. a + 4b – 8 = 0)
3. [M1] for attempt to substitute x = 2 or x = -1 into f(x) and use f(2) = 2f(-1) or 2f(2) = f(-1)
4. [A1] for a correct equation in any form (e.g. 10a + 4b + 10 = 0)
5. [DM1] (on both previous M marks) for attempt to solve simultaneous equations to obtain either a or b
6. [A1] for both correct (a = -2, b = 5/2)