233. If $x_{1}$ and $x_{2}$ are the roots of the equation

$$
\frac{3 a-b}{c} x^{2}+\frac{c(3 a+b)}{3 a-b}=0
$$

then, without solving it, find $x_{1}^{117}+x_{2}^{117}$.
Solution.

$x_{1}^{117}+x_{2}^{117}=\left(x_{1}+x_{2}\right)\left(x_{1}^{116}-x_{1}^{115} x_{2}+\ldots+x_{2}^{116}\right), \quad$ but $\quad x_{1}+x_{2}=0$, so, $x_{1}^{117}+x_{2}^{117}=0$.

Note. It is useful to solve problem 233 when reviewing, after discussing the equality:

$$
a^{n}+b^{n}=(a+b)\left(a^{n-1}-a^{n-2} b+\ldots+b^{n-1}\right)
$$[^5]

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