6 The equation

$$x ^ { 3 } - x + 1 = 0$$

has roots $\alpha , \beta , \gamma$.\\
(i) 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 $S _ { n } = \alpha ^ { n } + \beta ^ { n } + \gamma ^ { n }$.\\
(ii) Find the value of $S _ { - 3 }$.\\

(iii) Show that $S _ { 6 } = 5$ and find the value of $S _ { 9 }$.\\

\hfill \mbox{\textit{CAIE FP1 2019 Q6}}