Do not use a calculator in answering this question. One root of the equation \(2{{z}^{3}}-5{{z}^{2}}

Question

Do not use a calculator in answering this question. One root of the equation $2{{z}^{3}}-5{{z}^{2}}+\alpha z-5=0$, where $\alpha \in \mathbb{R}$, is $z=1-2\mathbf{i}$. Find the value of $\alpha $ and the other roots.[5]

Accepted Answer

$\alpha=12$, other roots $1+2\mathrm{i}$ and $\dfrac12$

2023 RI Promo Q8

2023 RI Promo Q8

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