Unclear math contest solution I'm having difficulty understanding this solution to a problem in an old math contest. I understand everything up to the word "Hence". Can anybody explain this to me?
Question

Answer


The three roots satisfy the polynomial.  So, from 
$$ x^5 = \cdots = -3 + 9x - x^2  $$
we get \begin{align*}
a^5 &= -3 + 9a - a^2  \\
b^5 &= -3 + 9b - b^2  \\
c^5 &= -3 + 9c - c^2  \text{.}
\end{align*}
Summing these, we get 
$$ a^5 + b^5 + c^5 = 3(-3) + 9(a+b+c) - (a^2 + b^2 + c^2) = \dots  \text{.}  $$
