2 The cubic equation
$$z ^ { 3 } + p z ^ { 2 } + 6 z + q = 0$$
has roots \(\alpha , \beta\) and \(\gamma\).
- Write down the value of \(\alpha \beta + \beta \gamma + \gamma \alpha\).
- Given that \(p\) and \(q\) are real and that \(\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } = - 12\) :
- explain why the cubic equation has two non-real roots and one real root;
- find the value of \(p\).
- One root of the cubic equation is \(- 1 + 3 \mathrm { i }\).
Find:
- the other two roots;
- the value of \(q\).