<s>[INST] Let $a,b,c$ be the roots of $x^3 - x - 1= 0$ find $a^5 + b^5 + c^5$ Consider a cubic polynomial $x^3 - x - 1 = 0$ I want the sum of the fifth powers of the roots $\sum a^5$.  I know that
\begin{eqnarray*}
a + b + c &=& 0 \\
ab + bc + ca &=& 1 \\
-abc &=& 1
\end{eqnarray*}
but I have no way of combining this information into the answer.
 [/INST] $a^3=a+1$ implies that $a^5=a^3+a^2=a+1+a^2$.
$a^5+b^5+c^5=3+a+b+c+a^2+b^2+c^2=3+(a+b+c)^2-2(ab+ac+bc)=5$.
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