<s>[INST] 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.
 [/INST] 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.
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