3 The cubic equation
$$z ^ { 3 } + q z + ( 18 - 12 i ) = 0$$
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 \(\beta + \gamma = 2\), find the value of:
- \(\alpha\);
- \(\quad \beta \gamma\);
- \(q\).
- Given that \(\beta\) is of the form \(k \mathrm { i }\), where \(k\) is real, find \(\beta\) and \(\gamma\).