(a) Let $n \geq 1$ be an integer. Prove that $X ^ { n } + Y ^ { n } + Z ^ { n }$ can be written as a polynomial with integer coefficients in the variables $\alpha = X + Y + Z$, $\beta = X Y + Y Z + Z X$ and $\gamma = X Y Z$. (b) Let $G _ { n } = x ^ { n } \sin ( n A ) + y ^ { n } \sin ( n B ) + z ^ { n } \sin ( n C )$, where $x , y , z , A , B , C$ are real numbers such that $A + B + C$ is an integral multiple of $\pi$. Using (a) or otherwise, show that if $G _ { 1 } = G _ { 2 } = 0$, then $G _ { n } = 0$ for all positive integers $n$.
(a) Let $n \geq 1$ be an integer. Prove that $X ^ { n } + Y ^ { n } + Z ^ { n }$ can be written as a polynomial with integer coefficients in the variables $\alpha = X + Y + Z$, $\beta = X Y + Y Z + Z X$ and $\gamma = X Y Z$.
(b) Let $G _ { n } = x ^ { n } \sin ( n A ) + y ^ { n } \sin ( n B ) + z ^ { n } \sin ( n C )$, where $x , y , z , A , B , C$ are real numbers such that $A + B + C$ is an integral multiple of $\pi$. Using (a) or otherwise, show that if $G _ { 1 } = G _ { 2 } = 0$, then $G _ { n } = 0$ for all positive integers $n$.