2017 NYJC P1 Q3

Do not use a calculator in answering this question.

1. Explain why the equation \({{z}^{3}}+a{{z}^{2}}+az+7=0\) cannot have more than two non-real roots, where \(a\) is a real constant.[1]
2. Given that \(z=-7\) is a root of the equation in (i), find the other roots, leaving your answers in the form \(r{{e}^{\mathrm{i}\theta }}\), where \(r>0\) and \(-\pi <\theta \le \pi \) .[4]
3. Hence, solve the equation \(\mathrm{i}{{z}^{3}}+8{{z}^{2}}-8\mathrm{i}z-7=0\), leaving your answers in the form \(r{{e}^{\mathrm{i}\theta }}\), where \(r>0\) and \(-\pi <\theta \le \pi \) .[2]