How to factorize $f(n)=a^n+a±1$

Question

How to Factorize :

$$f(n)=a^n+a±1:n\in \mathbb{N}$$

my try:

$f(n)=a^n+a±1:n\in \mathbb{N} \to f(n)=a^n+a^{n-2}-a^{n-2}+a+1\\f(n)=a^{n-2}(a^2+a^{3-n})-a^{n-2}+1!!!!:($

Answer 1

In general you cannot. For example the polynomial $x^5-x+1$ is not factorizable. See Abel- Ruffini theorem which says there is no resolution formula by radicals to solve a n-th degree polynomial for$\ 5\leq n $.

Answer 2

There is factorization of $a^n+a+1$ for all $n=3k+2$, where $k\in\mathbb N$ and of $a^n+a-1$ for all $n=3k+2$, where $k\in\mathbb 2N-1$.

Because we have $a^3+1=(a^2-a+1)(a+1)$ and $a^3-1=(a^2+a+1)(a-1)$. For example, $$a^5+a-1=a^5+a^2-a^2+a-1=(a^2-a+1)(a^3+a^2-1)$$