7 The equation $x ^ { 3 } + 2 x ^ { 2 } + x + 7 = 0$ has roots $\alpha , \beta , \gamma$.\\
(i) Use the relation $x ^ { 2 } = - 7 y$ to show that the equation

$$49 y ^ { 3 } + 14 y ^ { 2 } - 27 y + 7 = 0$$

has roots $\frac { \alpha } { \beta \gamma } , \frac { \beta } { \gamma \alpha } , \frac { \gamma } { \alpha \beta }$.\\

(ii) Show that $\frac { \alpha ^ { 2 } } { \beta ^ { 2 } \gamma ^ { 2 } } + \frac { \beta ^ { 2 } } { \gamma ^ { 2 } \alpha ^ { 2 } } + \frac { \gamma ^ { 2 } } { \alpha ^ { 2 } \beta ^ { 2 } } = \frac { 58 } { 49 }$.\\

(iii) Find the exact value of $\frac { \alpha ^ { 3 } } { \beta ^ { 3 } \gamma ^ { 3 } } + \frac { \beta ^ { 3 } } { \gamma ^ { 3 } \alpha ^ { 3 } } + \frac { \gamma ^ { 3 } } { \alpha ^ { 3 } \beta ^ { 3 } }$.\\

\hfill \mbox{\textit{CAIE FP1 2019 Q7}}