Problem

$f(z) = 6z^3 + Az^2 + Bz + C$

where $A, B$ and $C$ are integers.

Given that $z = \frac{2}{3} + \frac{\sqrt{17}}{3}\mathrm{i}$ is a root of the equation $f(z) = 0$

(a) write down the other complex root of the equation $f(z) = 0$

Given that $z = -\frac{3}{2}$ is also a root of the equation $f(z) = 0$

(b) determine the value of $A$, the value of $B$ and the value of $C$.

(c) Show all the roots of the equation $f(z) = 0$ on a single Argand diagram.