Polynomial $x^{8} + x^{7} + x^{6} + x^{5} + x^{4} + x^{3} + x^{2} + x + 1$ is reducible over $\mathbb{Q}$? Clearly, there are no roots, but how can I find factors of higher degree?

This is
$$(x^6+x^3+1)(x^2+x+1).$$
In fact a polynomial
$$\sum_{j=0}^m x^j=x^m+x^{m-1}+\cdots+x+1$$
for $m\ge1$
is irreducible over $\Bbb Q$ iff $m+1$ is prime.
