3.
\section*{In this question you must show detailed reasoning.}
You are given that \(\mathrm { f } ( z ) = 4 z ^ { 4 } - 12 z ^ { 3 } + 41 z ^ { 2 } - 128 z + 185\) and that \(2 + \mathrm { i }\) is a root of the equation \(\mathrm { f } ( z ) = 0\).
- Express \(\mathrm { f } ( z )\) as the product of two quadratic factors with integer coefficients.
- Solve \(\mathrm { f } ( z ) = 0\).
Two loci on an Argand diagram are defined by \(C _ { 1 } = \left\{ z : | z | = r _ { 1 } \right\}\) and \(C _ { 2 } = \left\{ z : | z | = r _ { 2 } \right\}\) where \(r _ { 1 } > r _ { 2 }\). You are given that two of the points representing the roots of \(\mathrm { f } ( z ) = 0\) are on \(C _ { 1 }\) and two are on \(C _ { 2 } . R\) is the region on the Argand diagram between \(C _ { 1 }\) and \(C _ { 2 }\).
- Find the exact area of \(R\).
- \(\omega\) is the sum of all the roots of \(\mathrm { f } ( z ) = 0\).
Determine whether or not the point on the Argand diagram which represents \(\omega\) lies in \(R\).