The polynomial p(x) is defined by p(x) = x² + ax + b, where a and b are constants. It is given that (x + 2) is a factor of p(x) and that the remainder is 5 when p(x) is divided by (x – 3).

Mark scheme

Marking points from the official mark scheme (Feb/Mar 2021). The mark or mark type is shown in brackets.

1. [M1*] Substitute x = -2 and equate to zero
2. [M1*] Substitute x = 3 and equate to 5
3. [A1] Obtain -8-2a+b=0 and 27 + 3a + b = 5 or equivalents
4. [DM1] Solve a pair of relevant linear simultaneous equations for a or b
5. [A1] Obtain a = -6 and b = -4
6. [M1] Attempt division by x + 2 at least as far as x² + kx
7. [A1] Obtain x²-2x-2
8. [A1] Obtain (at least) the positive root (2+√12)/2 or exact equivalent
9. [M1] Equate e²ʸ to positive root, apply logarithms and use power law
10. [A1] Obtain 1/2 ln((2+√12)/2) or 1/2 ln(1+√3) or exact equivalent