6.
$$\mathrm { f } ( z ) = 8 z ^ { 3 } + 12 z ^ { 2 } + 6 z + 65$$
Given that \(\frac { 1 } { 2 } - \mathrm { i } \sqrt { 3 }\) is a root of the equation \(\mathrm { f } ( z ) = 0\)
- write down the other complex root of the equation,
- use algebra to solve the equation \(\mathrm { f } ( z ) = 0\) completely.
- Show the roots of \(\mathrm { f } ( z )\) on a single Argand diagram.
- Show that the roots of \(\mathrm { f } ( z )\) form the vertices of an equilateral triangle in the complex plane.
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