Sum of fifth power of the roots of equation $x^3-x^2+1=0$ 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: 
$x^3=x^2-1$
$\alpha + \beta + \gamma = 1$
$\alpha \cdot \beta \cdot \gamma = -1$
$\alpha \cdot \beta + \beta \cdot \gamma + \alpha \cdot \gamma = 0$
So, $\alpha^3=\alpha^2-1$
$\alpha^5=\alpha^4-\alpha^2$
And similarly for $\beta$ and $\gamma$
Now I did add them but I am unable to find something useful in it.

$$\alpha^5+\beta^5+\gamma^5=(\alpha+\beta+\gamma)^5-5(\alpha+\beta+\gamma)^3(\alpha\beta+\alpha\gamma+\beta\gamma)+5(\alpha\beta+\alpha\gamma+\beta\gamma)^2+$$
$$+5(\alpha+\beta+\gamma)^2\alpha\beta\gamma-5(\alpha\beta+\alpha\gamma+\beta\gamma)\alpha\beta\gamma=1+5\cdot(-1)=-4.$$
