Factorise polynomial $x^{5n} + x^n + 1$

Question

I am so sorry if this problem will be a dublicate of something, but I looked for it everywhere - still couldn't find it. So there it goes:
Factorise $x^{5n} + x^n + 1$. I have already found that $x^5+x+1=(x^3-x^2+1)(x^2+x+1)$, but have no clue how could I use it. Thank you for any help you could provide me!

Answer 1

If $x^5+x+1=(x^3-x^2+1)(x^2+x+1)$, then\begin{align}x^{5n}+x^n+1&=(x^n)^5+x^n+1\\&=((x^n)^3-(x^n)^2+1)((x^n)^2+x^n+1)\\&=(x^{3n}-x^{2n}+1)(x^{2n}+x^n+1)\end{align}

Answer 2

For starters, you can use $(x^{3n}-x^{2n}+1)(x^{2n}+x^n+1)$.