Polynomial-related manipulation My question is:

Factorize: $$x^{11} + x^{10} + x^9 + \cdots + x + 1$$

Any help to solve this question would be greatly appreciated.

Since $x^{11}+x^{10}+\ldots + x+1 = \frac{x^{12}-1}{x-1}$ we may first factorize $x^{12}-1$ and then divide by the factor $x-1$:
\begin{align*}
x^{12}-1 &= (x^6-1)(x^6+1)\\
 &= (x^3-1)(x^3+1)(x^6+1)\\
 &=(x-1)(x^2+x+1)(x+1)(x^2-x+1)(x^2+1)(x^4-x^2+1),
\end{align*}
hence
$$x^{11}+x^{10}+\ldots +x+1 = (x^2+x+1)(x+1)(x^2-x+1)(x^2+1)(x^4-x^2+1).$$
It is an easy exercise to show that the factors are irreducible over $\mathbb Q$.
In fact, the factors are the cyclotomic polynomials of the divisors of 12 (except 1).
