4 The cubic equation
$$z ^ { 3 } + \mathrm { i } z ^ { 2 } + 3 z - ( 1 + \mathrm { i } ) = 0$$
has roots \(\alpha , \beta\) and \(\gamma\).
- Write down the value of:
- \(\alpha + \beta + \gamma\);
- \(\alpha \beta + \beta \gamma + \gamma \alpha\);
- \(\alpha \beta \gamma\).
- Find the value of:
- \(\alpha ^ { 2 } + \beta ^ { 2 } + \gamma ^ { 2 }\);
- \(\alpha ^ { 2 } \beta ^ { 2 } + \beta ^ { 2 } \gamma ^ { 2 } + \gamma ^ { 2 } \alpha ^ { 2 }\);
- \(\alpha ^ { 2 } \beta ^ { 2 } \gamma ^ { 2 }\).
- Hence write down a cubic equation whose roots are \(\alpha ^ { 2 } , \beta ^ { 2 }\) and \(\gamma ^ { 2 }\).