6 The equation
$$x ^ { 3 } - x + 1 = 0$$
has roots \(\alpha , \beta , \gamma\).
- Use the relation \(x = y ^ { \frac { 1 } { 3 } }\) to show that the equation
$$y ^ { 3 } + 3 y ^ { 2 } + 2 y + 1 = 0$$
has roots \(\alpha ^ { 3 } , \beta ^ { 3 } , \gamma ^ { 3 }\). Hence write down the value of \(\alpha ^ { 3 } + \beta ^ { 3 } + \gamma ^ { 3 }\).
Let \(S _ { n } = \alpha ^ { n } + \beta ^ { n } + \gamma ^ { n }\). - Find the value of \(S _ { - 3 }\).
- Show that \(S _ { 6 } = 5\) and find the value of \(S _ { 9 }\).