N2017 P1 Q8

Do not use a calculator in answering this question.

1. Find the roots of the equation \({{z}^{2}}\left( 1-\mathbf{i} \right)-2z+\left( 5+5\,\mathbf{i} \right)=0\), giving your answers in cartesian form \(a+\mathbf{i}\,b\).[3]
2. 1. Given that \(\omega =1-\mathbf{i}\), find \({{\omega }^{2}}\), \({{\omega }^{3}}\) and \({{\omega }^{4}}\) in cartesian form. Given also that \({{\omega }^{4}}+p{{\omega }^{3}}+39{{\omega }^{2}}+q\omega +58=0\),where \(p\) and \(q\) are real, find \(p\) and \(q\).[4]
   2. Using the values of \(p\) and \(q\) in part (b)(i), express \({{\omega }^{4}}+p{{\omega }^{3}}+39{{\omega }^{2}}+q\omega +58\) as the product of two quadratic factors.[3]