If α and β are the roots of x^2+x+1=0 , then what is ∑_ j=0 ^3 (...

$\because x^2+x+1=0\Rightarrow \alpha=\omega$ and $\beta=\omega^2\therefore \sum_{j=0}^3$\left(\alpha^j+\beta^j\right)=\left(\alpha^0+\alpha^1+\alpha^2+\alpha^3\right)+\left(\beta^0+\beta^1+\beta^2+\beta^3\right)\because$sum of 3 consecutive powers of$\alpha$and$\beta=0\therefore \sum_{j=0}^3\left(\alpha^j+\beta^j\right)=\alpha^0+\beta^0=2