Do not use a calculator in answering this question. Given that $-\mathbf{i}$ is a root of the equati

Question

Do not use a calculator in answering this question. Given that $-\mathbf{i}$ is a root of the equation ${{z}^{3}}+k{{z}^{2}}+\left( 8+2\sqrt{2}\,\mathbf{i} \right)z+8\mathbf{i}=0$, where $k$ is a constant to be determined, find the other roots, leaving your answers in exact cartesian form $x+y\mathbf{i}$, showing your working. [3] Hence solve the equation $\mathbf{i}{{z}^{3}}+k{{z}^{2}}+\left( 2\sqrt{2}-8\mathbf{i} \right)z-8\mathbf{i}=0$, leaving your answers in exact cartesian form. [2]

Accepted Answer

(i) $k=2\sqrt{2}+\mathbf{i}$, $-\sqrt{2}+\sqrt{6}\mathbf{i}$ and $-\sqrt{2}-\sqrt{6}\mathbf{i}$ (ii) $z=-1$, $-\sqrt{6}+\sqrt{2}\mathbf{i}$, 	ext{ or } $\sqrt{6}+\sqrt{2}\mathbf{i}$

RI Complex Numbers Tutorial Q7

RI Complex Numbers Tutorial Q7

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