f(x) = x³ + px² + qx + 6 Given that (x + 1) is a factor of f(x) and that when f(x) is divided by (x - 2) the remainder is 30.

Mark scheme

Marking points from the official mark scheme. The mark or mark type is shown in brackets.

1. (a) M1 for using the Factor Theorem, f(-1) = 0.
2. (a) A1 for substituting x=-1 into f(x) to get (-1)³ + p(-1)² + q(-1) + 6 = 0 or -1 + p - q + 6 = 0.
3. (a) A1 for simplifying to p - q = -5 (cao).
4. (b) M1 for using the Remainder Theorem, f(2) = 30.
5. (b) A1 for substituting x=2 into f(x) and setting equal to 30: 2³ + p(2)² + q(2) + 6 = 30, leading to 4p + 2q = 16 or 2p + q = 8.
6. (b) M1 for attempting to solve the two simultaneous equations (p - q = -5 and 2p + q = 8).
7. (b) A1 for p = 1 and q = 6.
8. (c) M1 for correctly performing algebraic division (or synthetic division) of f(x) by (x+1) using their p and q to find a quadratic factor.
9. (c) A1 for obtaining the quadratic factor (x² + 6).
10. (c) M1 for attempting to factorise the quadratic factor (if possible) or stating that it cannot be factorised further over real numbers (e.g., checking discriminant).
11. (c) A1 for f(x) = (x + 1)(x² + 6) only (or x+1 and x²+6 explicitly stated as factors).