Factorize $a^5+a^4+a^3+a^2+a+1$

I have to factorize $a^5+a^4+a^3+a^2+a+1$. I tried to give rewrite it as\begin{align}a^4(a + 1) + a^2(a + 1) + 1 &= (a + 1)(a^4+a^2) + 1\\&=(a + 1)a^2(a^2 + 1)+1,\end{align}but i couldn't get anywhere from here. I also tried writing it as as\begin{align}a^5+a^3+a^4+2a^2+1-a^2 &= a^3(a^2+1) + (a^2+1)^2 - a^2 \\&= (a^2+1)(a^3+a^2+1)-a^2.\end{align}It seems that i am not getting anywhere with my approach. Can someone help me in finding a solution?

Solution 1:

You have\begin{align}a^5+a^4+a^3+a^2+a+1&=\frac{a^6-1}{a-1}\\&=\frac{(a^3-1)(a^3+1)}{a-1}\\&=(a^2+a+1)(a^3+1)\\&=(a^2+a+1)(a+1)(a^2-a+1).\end{align}