Problem

$f(z) = 2z^3 + pz^2 + qz – 41$

where $p$ and $q$ are integers.

The complex number $5 – 4i$ is a root of the equation $f(z) = 0$

(a) Write down another complex root of this equation.

(b) Solve the equation $f(z) = 0$ completely.

(c) Determine the value of $p$ and the value of $q$.

When plotted on an Argand diagram, the points representing the roots of the equation $f(z) = 0$ form the vertices of a triangle.

(d) Determine the area of this triangle.