6 The equation
$$x ^ { 3 } + x - 1 = 0$$
has roots \(\alpha , \beta , \gamma\). Use the relation \(x = \sqrt { } y\) to show that the equation
$$y ^ { 3 } + 2 y ^ { 2 } + y - 1 = 0$$
has roots \(\alpha ^ { 2 } , \beta ^ { 2 } , \gamma ^ { 2 }\).
Let \(S _ { n } = \alpha ^ { n } + \beta ^ { n } + \gamma ^ { n }\).
- Write down the value of \(S _ { 2 }\) and show that \(S _ { 4 } = 2\).
- Find the values of \(S _ { 6 }\) and \(S _ { 8 }\).