9 The roots of the equation \(x ^ { 3 } - k x ^ { 2 } - 2 = 0\) are \(\alpha , \beta\) and \(\gamma\), where \(\alpha\) is real and \(\beta\) and \(\gamma\) are complex.
- Show that \(k = \alpha - \frac { 2 } { \alpha ^ { 2 } }\).
- Given that \(\beta = u + \mathrm { i } v\), where \(u\) and \(v\) are real, find \(u\) in terms of \(\alpha\).
- Find \(v ^ { 2 }\) in terms of \(\alpha\).