2020 HCI P1 Q8

Do not use a graphing calculator in answering this question.

The complex number \(w\) is a root to the equation

\({{z}^{4}}+a{{z}^{3}}+46{{z}^{2}}+bz+125=0\),

where \(a\),\(b\in \mathbb{R}\)

1. Prove that the product of the factors \(z-w\) and \(z-{{w}^{*}}\) is a quadratic polynomial with real coefficients.[2]
2. Given that \(\left| w \right|=\sqrt{5}\) and \(\arg (w)=-{{\tan }^{-1}}(2)\) where \(-\frac{\pi }{2}<-{{\tan }^{-1}}(2)<0\), show that \(w=1-2\mathrm{i}\).[2]
3. Find all the roots of the equation and the values \(a\) and \(b\).[5]