Problem

3. $f(z) = z^4 + az^3 + bz^2 + cz + d$

where $a$, $b$, $c$ and $d$ are integers.

The complex numbers $3 + i$ and $-1 - 2i$ are roots of the equation $f(z) = 0$

(a) Write down the other roots of this equation.

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

(c) Determine the values of $a$, $b$, $c$ and $d$.