4 The roots of the cubic equation
$$z ^ { 3 } - 2 z ^ { 2 } + p z + 10 = 0$$
are \(\alpha , \beta\) and \(\gamma\).
It is given that \(\alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 } = - 4\).
- Write down the value of \(\alpha + \beta + \gamma\).
- Explain why \(\alpha ^ { 3 } - 2 \alpha ^ { 2 } + p \alpha + 10 = 0\).
- Hence show that
$$\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 } = p + 13$$
- Deduce that \(p = - 3\).
- Find the real root \(\alpha\) of the cubic equation \(z ^ { 3 } - 2 z ^ { 2 } - 3 z + 10 = 0\).
- Find the values of \(\beta\) and \(\gamma\).