2.
$$f ( z ) = 2 z ^ { 3 } + p z ^ { 2 } + q z - 41$$
where \(p\) and \(q\) are integers.
The complex number \(5 - 4 \mathrm { i }\) is a root of the equation \(\mathrm { f } ( \mathrm { z } ) = 0\)
- Write down another complex root of this equation.
- Solve the equation \(\mathrm { f } ( \mathrm { z } ) = 0\) completely.
- Determine the value of \(p\) and the value of \(q\).
When plotted on an Argand diagram, the points representing the roots of the equation \(\mathrm { f } ( \mathrm { z } ) = 0\) form the vertices of a triangle.
- Determine the area of this triangle.