<s>[INST] Factoring $x^5 + x^4 + x^3 + x^2 + x + 1$ without using $\frac{x^n - 1}{x-1}$? I was at a math team meet today and one of the problems was to factor $x^5 + x^4 + x^3 + x^2 + x + 1$. It also gave the hint that it decomposes into two trinomials and a binomial.
The solution they gave was based on the fact that $\frac{x^6 - 1}{x-1} = x^5 + x^4 + x^3 + x^2 + x + 1$ and from there the solution is pretty straightforward. However, I was not aware of that factorization. The only ones I have really learned are $x^2 - y^2 = (x-y)(x+y)$ and $x^3 \pm y^3 = (x \pm y)(x^2 \mp xy + y^2)$. Is there any other way I could have solved this factorization without using the ones they used?
 [/INST] You should be(come) aware of:  $\frac{x^n-1}{x-1} = \sum_{k=0}^{n-1} x^k$
It's quite useful.
$$x^5+x^4+x^3+x^2+x+1 \\ = \frac{x^6-1}{x-1} \\ = \frac{(x^3-1)(x^3+1)}{x-1} \\ = (x^2+x+1)(x^3+1) \\ = (x^2+x+1)(x+1)(x^2-x+1)$$
The only other way would be to guess -1 as a root, because you have six terms in ascending polynomial sequence. 
$$ x^5+x^4+x^3+x^2+x+1 = (x+1)(x^4+x^2+1)$$
Then split the resulting tetranomial, by solving: $$\exists a, b : x^4 + x^2 + 1 = (x^2+ax+1)(x^2+bx+1) \\ (x^2+a x + 1)(x^2+b x +1) = x^4 + (a+b) x^3 + (2+ab) x^2 + (a+b)x + 1 \\ \therefore a+b=0 \land ab=-1 \\ \therefore a = \pm 1, b=\mp 1 $$
And so:
$$ x^5+x^4+x^3+x^2+x+1 = (x+1)(x^2+x+1)(x^2-x+1)$$
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