4 The cubic equation
$$z ^ { 3 } + p z + q = 0$$
has roots \(\alpha , \beta\) and \(\gamma\).
- Write down the value of \(\alpha + \beta + \gamma\).
- Express \(\alpha \beta \gamma\) in terms of \(q\).
- Show that
$$\alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 } = 3 \alpha \beta \gamma$$
- Given that \(\alpha = 4 + 7 \mathrm { i }\) and that \(p\) and \(q\) are real, find the values of:
- \(\beta\) and \(\gamma\);
- \(p\) and \(q\).
- Find a cubic equation with integer coefficients which has roots \(\frac { 1 } { \alpha } , \frac { 1 } { \beta }\) and \(\frac { 1 } { \gamma }\).