Factoristaion of $x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+1$ My question revolves around the polynomial
$$x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+1$$
I know that it can be factorised into
$$(x^2+x+1)(x^6+x^3+1)$$
but what method can we employ to obtain this result in the first place? Any method that works will be of great help to me.
Thank you for your help.

Method 1: Use what you know about polynomials of the form $x^n-1$. For instance, your polynomial is $\frac{x^9-1}{x-1}$. So if you can factor the numerator, many of those factors would be inherited to your polynomial. And indeed, we have
$$
x^9-1=(x^3)^3-1=((x^3)^2+x^3+1)(x^3-1)\\
=(x^6+x^3+1)(x^2+x+1)(x-1)
$$
Put this into the fraction above, and the factorisation follows.
Method 2: Inspection. There are nine terms. They can be grouped three-by-three:
$$
(x^8+x^7+x^6)+(x^5+x^4+x^3)+(x^2+x+1)\\
=x^6(x^2+x+1)+x^3(x^2+x+1)+(x^2+x+1)
$$
and the factorisation follows.
