Roots of a quadratic equation If $ a $ , $b$ are the roots of $ x^2+x+1=0 $ , then the equation whose roots are $a^k $ and $b ^k$ where k is a positive integer not divisible by 3 is 
$a)$ $x^2 - x + 1 = 0$
$b)$ $x^2 + x+1 = 0$
$c)$ $ x^2 -x -1 =0$
$d)$ None of the above 
My attempt : I was able to solve the question , only after I used the options given below . I put k = 2 and checked the options for which the above statement was coming to be true.Using the formula for $(a+b)^2$ .
However I am looking for a more genera approach , preferably one which doesn't apply checking against the options

Hint: the roots are $$\omega,\omega^2$$ so sum of roots of the wanted quadratic is $\omega^k+\omega^{2k}=(-1)$ as $3|k$ (excluded) and product is equal to $\omega^{2k+k}=1$ thus the correct option is $x^2+x+1$.
