2022 ASRJC P2 Q1
(i)
The equation $3{{z}^{3}}-7{{z}^{2}}+17z+m=0$, where $m$ is a real constant, has a root $z=1+2\mathbf{i}$. Find the value of $m$.
Hence using an algebraic method, find all the roots of the equation $3{{z}^{3}}-7{{z}^{2}}+17z+m=0$. Show your working clearly.
[4]
(ii)
Hence, solve the equation $\frac{3}{{{w}^{3}}}+\frac{7}{{{w}^{2}}}+\frac{17}{w}-m=0$, giving your answers in the form $a+b\mathbf{i}$, where $a$, $b\in \mathbb{R}$.
[2]
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