The polynomial p(x) is defined by p(x) = ax³ + bx² – 17x – a, where a and b are constants. It is given that (x + 2) is a factor of p(x) and that the remainder is 28 when p(x) is divided by (x – 2).

Mark scheme

Marking points from the official mark scheme (Feb/Mar 2017). The mark or mark type is shown in brackets.

1. [M1] Substitute x = −2 and equate to zero
2. [M1] Substitute x = 2 and equate to 28
3. [A1] Obtain −9a + 4b + 34 = 0 and 7a + 4b – 62 = 0 or equivalents
4. [M1] Solve a relevant pair of simultaneous equations for a or b
5. [A1] Obtain a = 6, b = 5
6. [M1] Divide by x + 2, or equivalent, at least as far as k₁x² + k₂x
7. [A1] Obtain 6x² – 7x – 3
8. [A1] Obtain (x + 2)(3x + 1)(2x – 3)
9. [M1] Refer to, or clearly imply, fact that 2ˣ is positive
10. [A1] State one