4 The roots of the equation
$$z ^ { 3 } - 5 z ^ { 2 } + k z - 4 = 0$$
are \(\alpha , \beta\) and \(\gamma\).
- Write down the value of \(\alpha + \beta + \gamma\) and the value of \(\alpha \beta \gamma\).
- Hence find the value of \(\alpha ^ { 2 } \beta \gamma + \alpha \beta ^ { 2 } \gamma + \alpha \beta \gamma ^ { 2 }\).
- The value of \(\alpha ^ { 2 } \beta ^ { 2 } + \beta ^ { 2 } \gamma ^ { 2 } + \gamma ^ { 2 } \alpha ^ { 2 }\) is - 4 .
- Explain why \(\alpha , \beta\) and \(\gamma\) cannot all be real.
- By considering \(( \alpha \beta + \beta \gamma + \gamma \alpha ) ^ { 2 }\), find the possible values of \(k\).