2022 ACJC P2 Q3
(i)
Find the roots of the equation $\mathbf{i}{{z}^{2}}-\left( 5+\mathbf{i} \right)z+2-6\mathbf{i}=0$, giving your answers in cartesian form $a+b\mathbf{i}$, where $a$, $b\in \mathbb{R}$.
[2]
(ii)
Hence find the roots of the equation $-\mathbf{i}{{w}^{2}}-\left( 1-5\mathbf{i} \right)w+2-6\mathbf{i}=0$, giving your answers in cartesian form $a+b\mathbf{i}$, where $a$, $b\in \mathbb{R}$.
[2]
(iii)
Given that the roots found in part (i) are also roots of the equation $P\left( z \right)=0$, where $P\left( z \right)$ is a polynomial of degree $4$ with real coefficients, find $P\left( z \right)$.
[3]
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