Polynomial factors Why must $x^2 + x + 1$ be a factor of  $x^5+x^4+x^3+x^2+x+1$?
I know that when we divide  $x^5+x^4+x^3+x^2+x+1$ by $x^3+1$ we get $x^2 + x + 1$, but is there an argument/theorem or anything that could tell that $x^2+x+1$ must divide $x^2 + x + 1$?

It depends on what you count as telling. It follows from an easy, basic manipulation of familiar identities, but I don’t see any way to recognize this without appealing to something at least somewhat computational. I look at the sum $x^5+x^4+x^3+x^2+x+1$ and immediately think of $x^6-1$, and from there it all falls out:
$$\begin{align*}
x^5+x^4+x^3+x^2+x+1&=\frac{x^6-1}{x-1}\\
&=\frac{(x^3)^2-1}{x-1}\\
&=\frac{(x^3-1)(x^3+1)}{x-1}\\
&=(x^3+1)\frac{x^3-1}{x-1}\\
&=(x^3+1)(x^2+x+1)\;,
\end{align*}$$
all using standard factorizations.
