2021 TMJC P1 Q4
A quartic (degree four) polynomial $\text{P}\left( z \right)={{z}^{4}}+a{{z}^{3}}+b{{z}^{2}}+cz+d$ has real coefficients. The equation $\text{P}\left( z \right)=0$ has root $r{{\text{e}}^{\mathbf{i}\theta }}$, where $r>0$ and $0<\theta <\pi $.
(i)
Write down a second root in terms of $r$ and $\theta $, and hence show that a quadratic factor of $\text{P}\left( z \right)$ is ${{z}^{2}}-\left( 2r\cos \theta \right)z+{{r}^{2}}$.
[3]
(ii)
Given that $\sqrt{3}{{\text{e}}^{\mathbf{i}\,\frac{\pi }{6}}}$ and $\sqrt{2}{{\text{e}}^{\mathbf{i}\,\frac{\pi }{4}}}$ are two roots of the equation $\text{P}\left( z \right)=0$, find an expression for $\text{P}\left( z \right)$ 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]
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