One root of the equation ${{z}^{4}}+2{{z}^{3}}+a{{z}^{2}}+bz+50=0$, where $a$ and $b$ are

Question

One root of the equation ${{z}^{4}}+2{{z}^{3}}+a{{z}^{2}}+bz+50=0$, where $a$ and $b$ are real, is $z=1+3\mathbf{i}$. Show that $a=7$ and $b=30$ and find the other roots of the equation.[5] Deduce the roots of the equation ${{w}^{4}}-2\mathbf{i}{{w}^{3}}-7{{w}^{2}}+30\mathbf{i}w+50=0$[2]

Accepted Answer

$\text{(i)}\ a=7,\ b=30$, with other roots $1-3\mathbf{i}$ and $-2\pm\mathbf{i}$ $\text{(ii)}\ w=3-\mathbf{i},\ -3-\mathbf{i},\ 1+2\mathbf{i},\ -1+2\mathbf{i}$

2017 RI P1 Q9

2017 RI P1 Q9

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