The equation $x^3-x^2+1=0$ has three roots $\alpha$, $\beta$ and $\gamma$. Find the value of $\alpha^5 + \beta^5 + \gamma^5$
I tried it this way: $x^3=x^2-1$
$\alpha + \beta + \gamma = 1$
$\alpha \cdot \beta \cdot \gamma = -1$
$\alpha \cdot \beta + \beta \cdot \gamma + \alpha \cdot \gamma = 0$
And similarly for $\beta$ and $\gamma$ Now I did add them but I am unable to find something useful in it.