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 $S _ { n } = \alpha ^ { n } + \beta ^ { n } + \gamma ^ { n }$.\\
(i) Write down the value of $S _ { 2 }$ and show that $S _ { 4 } = 2$.\\
(ii) Find the values of $S _ { 6 }$ and $S _ { 8 }$.

\hfill \mbox{\textit{CAIE FP1 2010 Q6}}