One of the roots of the equation ${{z}^{4}}-2{{z}^{3}}+a{{z}^{2}}-8z+40=0$ is $b\mathrm{i}$, where $

Question

One of the roots of the equation ${{z}^{4}}-2{{z}^{3}}+a{{z}^{2}}-8z+40=0$ is $b\mathrm{i}$, where $a$ and $b$ are positive real constants. Find the values of $a$ and $b$ and hence find the other roots of the equation. [4] Deduce that roots of the equation ${{w}^{4}}+2{{w}^{3}}+a{{w}^{2}}+8w+40=0$. [3]

Accepted Answer

(i) $a = 14$, $b = 2$; Roots: $2\mathrm{i}$, $-2\mathrm{i}$, $1+3\mathrm{i}$, $1-3\mathrm{i}$; (ii) $w = -2\mathrm{i}$, $2\mathrm{i}$, $-1-3\mathrm{i}$, $-1+3\mathrm{i}$

2019 MI P2 Q5 (a)

2019 MI P2 Q5 (a)

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