Verify that $z=\mathrm{i}$ is a root of the equation \({{z}^{4}}-2{{z}^{3}}+6{{z}^{2}}-2z+5=0

Question

Verify that $z=\mathrm{i}$ is a root of the equation ${{z}^{4}}-2{{z}^{3}}+6{{z}^{2}}-2z+5=0$.[1] Hence determine the other roots.[4]

Accepted Answer

${1+2\mathrm{i}}$, ${1-2\mathrm{i}}$ and ${-\mathrm{i}}$

N2005 P2 Q1