Problem

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

$f(z) = 4z^3 - pz^2 - 24z - 108$

where $p$ is a constant.

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

(a) determine the value of $p$

(b) using algebra, solve $f(z) = 0$ completely, giving the roots in simplest form,

(c) determine the modulus of the complex roots of $f(z) = 0$

(d) show the roots of $f(z) = 0$ on a single Argand diagram.