A graphic calculator is not to be used in answering this question. It is given that \({{z}_{1}}

Question

A graphic calculator is not to be used in answering this question. It is given that ${{z}_{1}}=1+\sqrt{3}\mathbf{i}$. Find the value of ${{z}_{1}}^{3}$, showing clearly how you obtain your answer. [3] Given that $1+\sqrt{3}\mathbf{i}$ is a root of the equation $2{{z}^{3}}+a{{z}^{2}}+bz+4=0$, find the values of the real numbers $a$ and $b$. [4] For these values of $a$ and $b$, solve the equation in part (ii), and show all the roots on an Argand diagram.[4]

Accepted Answer

(i) $-8$ (ii) $a=-3$, $b=6$ (iii) $1\pm \sqrt{3}\mathbf{i}$, $-\frac{1}{2}$