Do not use a calculator in answering this question. It is given that $1+2\mathrm{i}$ is a com

Question

Do not use a calculator in answering this question. It is given that $1+2\mathrm{i}$ is a complex root of the equation ${{z}^{3}}+p{{z}^{2}}+qz-10=0$, where $p$ and $q$ are real numbers. State another complex root of the equation in the form $a+\mathrm{i}b$, where $a$ and $b$ are real numbers and $b\ne 0$. Justify your answer.[2] Find the values of $p$ and $q$, and find the third root.[6] Hence find the roots of the equation $1+pw+q{{w}^{2}}-10{{w}^{3}}=0$.[2]

Accepted Answer

(a) $1 - 2\mathrm{i}$; (b) $p = -4$, $q = 9$, third root $= 2$; (c) $\displaystyle\frac{1-2\mathrm{i}}{5}$, $\displaystyle\frac{1+2\mathrm{i}}{5}$, $\displaystyle\frac{1}{2}$

N2025 P1 Q12

N2025 P1 Q12

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