2018 NYJC P2 Q3

Do not use a calculator in answering this question.

1. The equation \({{z}^{3}}+4{{z}^{2}}+8z+8=0\) has a root \(z=-2\). Find the other two roots \({{z}_{1}}\) and \({{z}_{2}}\) where \(-\pi <\arg {{z}_{2}}<\arg {{z}_{1}}\le \pi \).[2]
2. Find the modulus and argument of \(w\), where \(w=\frac{{{z}_{1}}}{{{z}_{2}}}\).[3]
3. Find the set of positive integers \(n\) for which \({{w}^{n}}\) is real, and show that, for these values of \(n\), \({{w}^{n}}\) is \(1\).[1]
4. Express \({{w}^{100}}-{{\left( {{w}^{*}} \right)}^{100}}\) in the form \(k\mathbf{i}\), giving the exact value of \(k\) in non-trigonometrical form.[2]