Find the roots of the equation $\mathbf{i}{{z}^{2}}-\left( 5+\mathbf{i} \right)z+2-6\mathbf{i}=0$,

Question

Find the roots of the equation $\mathbf{i}{{z}^{2}}-\left( 5+\mathbf{i} \right)z+2-6\mathbf{i}=0$, giving your answers in cartesian form $a+b\mathbf{i}$, where $a$, $b\in \mathbb{R}$.[2] Hence find the roots of the equation $-\mathbf{i}{{w}^{2}}-\left( 1-5\mathbf{i} \right)w+2-6\mathbf{i}=0$, giving your answers in cartesian form $a+b\mathbf{i}$, where $a$, $b\in \mathbb{R}$.[2] Given that the roots found in part (i) are also roots of the equation $P\left( z \right)=0$, where $P\left( z \right)$ is a polynomial of degree $4$ with real coefficients, find $P\left( z \right)$. [3]

Accepted Answer

(i) $z=1-3\mathrm{i}$ or $z=-2\mathrm{i}$ (ii) $w=3+\mathrm{i}$ or $w=2$ (iii) $P(z)=z^4-2z^3+14z^2-8z+40$

2022 ACJC P2 Q3

2022 ACJC P2 Q3

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