2 The cubic equation \(2 x ^ { 3 } - 4 x ^ { 2 } + 3 = 0\) has roots \(\alpha , \beta , \gamma\). Let \(\mathrm { S } _ { \mathrm { n } } = \alpha ^ { \mathrm { n } } + \beta ^ { \mathrm { n } } + \gamma ^ { \mathrm { n } }\).
- State the value of \(S _ { 1 }\) and find the value of \(S _ { 2 }\).
- Express \(\mathrm { S } _ { \mathrm { n } + 3 }\) in terms of \(\mathrm { S } _ { \mathrm { n } + 2 }\) and \(\mathrm { S } _ { \mathrm { n } }\).
- Hence, or otherwise, find the value of \(S _ { 4 }\).
- Use the substitution \(\mathrm { y } = \mathrm { S } _ { 1 } - \mathrm { x }\), where \(S _ { 1 }\) is the numerical value found in part (a), to find and simplify an equation whose roots are \(\alpha + \beta , \beta + \gamma , \gamma + \alpha\).
- Find the value of \(\frac { 1 } { \alpha + \beta } + \frac { 1 } { \beta + \gamma } + \frac { 1 } { \gamma + \alpha }\).