2022 HCI P1 Q8
Do not use a graphing calculator for this question.
(i)
The complex number $a+\mathbf{i}$ is a root of the equation ${{z}^{2}}-2\sqrt{2}z+b=0$ where $a$ and $b$ are positive real constants. Find the exact values of $a$ and $b$.
[3]
It is given that $\text{f}(z)={{z}^{6}}-3{{z}^{4}}+11{{z}^{2}}-9$.
(ii)
Show that $\text{f}(z)=\text{f}(-z)$.
[1]
(iii)
It is given that $a+\mathbf{i}$ is a root of the equation $\text{f}(z)=0$. Using the results found in parts (i) and (ii), show that $\text{f}(z)$ can be factorised into $3$ quadratic factors with real coefficients, leaving your answer in exact form.
[4]
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