5 The cubic equation
$$z ^ { 3 } - 4 \mathrm { i } z ^ { 2 } + q z - ( 4 - 2 \mathrm { i } ) = 0$$
where \(q\) is a complex number, has roots \(\alpha , \beta\) and \(\gamma\).
- Write down the value of:
- \(\alpha + \beta + \gamma\);
- \(\alpha \beta \gamma\).
- Given that \(\alpha = \beta + \gamma\), show that:
- \(\alpha = 2 \mathrm { i }\);
- \(\quad \beta \gamma = - ( 1 + 2 \mathrm { i } )\);
- \(\quad q = - ( 5 + 2 \mathrm { i } )\).
- Show that \(\beta\) and \(\gamma\) are the roots of the equation
$$z ^ { 2 } - 2 \mathrm { i } z - ( 1 + 2 \mathrm { i } ) = 0$$
- Given that \(\beta\) is real, find \(\beta\) and \(\gamma\).