2021 TMJC P1 Q4

A quartic (degree four) polynomial \(\mathrm{P}\left( z \right)={{z}^{4}}+a{{z}^{3}}+b{{z}^{2}}+cz+d\) has real coefficients. The equation \(\mathrm{P}\left( z \right)=0\) has root \(r{{\mathrm{e}}^{\mathbf{i}\theta }}\), where \(r>0\) and \(0<\theta <\pi \).

1. Write down a second root in terms of \(r\) and \(\theta \), and hence show that a quadratic factor of \(\mathrm{P}\left( z \right)\) is \({{z}^{2}}-\left( 2r\cos \theta \right)z+{{r}^{2}}\).[3]
2. Given that \(\sqrt{3}{{\mathrm{e}}^{\mathbf{i}\,\frac{\pi }{6}}}\) and \(\sqrt{2}{{\mathrm{e}}^{\mathbf{i}\,\frac{\pi }{4}}}\) are two roots of the equation \(\mathrm{P}\left( z \right)=0\), find an expression for \(\mathrm{P}\left( z \right)\) in the form \({{z}^{4}}+a{{z}^{3}}+b{{z}^{2}}+cz+d\) where \(a\), \(b\), \(c\) and \(d\) are constants to be determined.[4]