JAMB 1984 · UME · Q26

If $(x - 2)$ and $(x + 1)$ are factors of the expression $x^3 + px^2 + qx + 1$ , what is the sum of $p$ and $q$ ?

Worked solution (try it first)

By the factor theorem, the expression is 0 at $x = -1$ : $-1 + p - q + 1 = 0$ , so $p = q$ .
It is also 0 at $x = 2$ : $8 + 4p + 2q + 1 = 0$ , so $4p + 2q = -9$ .
Since $q = p$ , this is $6p = -9$ , so $p = q = -\frac32$ .
So $p + q = -3$ , option B.