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Q.
If $a, b, c$ are the roots of cubic equation $x^3-2 x+3=0$ and $f(x)=a^x+b^x+c^x$, then the value of $\frac{ f (10)+3 f (7)}{ f (8)}$ is equal to
Complex Numbers and Quadratic Equations
Solution:
$\Theta f (10) + f (7)= a ^{10}+ b ^{10}+ c ^{10}+3 a ^7+3 b ^7+3 c ^7 $
$= a ^7\left( a ^3+3\right)+ b ^7\left( b ^3+3\right)+ c ^7\left( c ^3+3\right)$
$ = a ^7 \cdot 2 a + b ^7 \cdot 2 b + c ^7 \cdot 2 c $
$ =2\left( a ^8+ b ^8+ c ^8\right)$
$\Rightarrow \frac{ f (10)+3 f (7)}{ f (8)}=2$