4 The cubic equation \(27 x ^ { 3 } + 18 x ^ { 2 } + 6 x - 1 = 0\) has roots \(\alpha , \beta , \gamma\).
- Show that a cubic equation with roots \(3 \alpha + 1,3 \beta + 1,3 \gamma + 1\) is
$$y ^ { 3 } - y ^ { 2 } + y - 2 = 0$$
The sum \(( 3 \alpha + 1 ) ^ { n } + ( 3 \beta + 1 ) ^ { n } + ( 3 \gamma + 1 ) ^ { n }\) is denoted by \(\mathrm { S } _ { \mathrm { n } }\).
- Find the values of \(S _ { 2 }\) and \(S _ { 3 }\).
- Find the values of \(S _ { - 1 }\) and \(S _ { - 2 }\).