Problem

In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

$f(z) = z^3 - 13z^2 + 59z - p \quad p \in \mathbb{Z}$

Given that $z = 3$ is a root of the equation $f(z) = 0$

(a) show that $p = -87$

(b) Use algebra to determine the other roots of $f(z) = 0$, giving your answers in simplest form.

On an Argand diagram

• the root $z = 3$ is represented by the point $P$
• the other roots of $f(z) = 0$ are represented by the points $Q$ and $R$
• the number $z = -9$ is represented by the point $S$

(c) Show on a single Argand diagram the positions of $P$, $Q$, $R$ and $S$

(d) Determine the perimeter of the quadrilateral $PQSR$, giving your answer as a simplified surd.