When $f(x) = {x^4} + 3{x^3} + p{x^2} - 2x + q$ is divided by (x − 2) the remainder is 15, and (x + 3) is a factor of f(x) .

Find the values of p and q .

$f(2) = 16 + 24 + 4p - 4 + q = 15$     M1

$\Rightarrow 4p + q = - 21$     A1

$f( - 3) = 81 - 81 + 9p + 6 + q = 0$     M1

$\Rightarrow 9p + q = - 6$     A1

$\Rightarrow p = 3{\text{ and }}q = - 33$     A1A1     N0

[6 marks]

Most candidates made a meaningful attempt at this question. Weaker candidates often made arithmetic errors and a few candidates tried using long division, which also often resulted in arithmetic errors. Overall there were many fully correct solutions.