Assume $f_n(x)=x^n+x^k+1$ is given, I will normally be interested in large $n.$
"On average" how does the degree of the highest degree irreducible divisor of $f_n$ compare to $n$? What, if anything is known? Are averages and moments known?
Motivation: Starting with an irreducible (usually primitive but irreducible is fine) polynomial $g(x)$ of degree $d,$ the goal is to determine the lowest degree trinomial multiple of $g(x)$. Call this degree $n=n(d).$ I am asking about the reverse problem above. The polynomial $g$ is used to generate LFSR sequences in cryptography, see Wikipedia.
If a brute force approach is adopted by examining $a(x) g(x)$ for $a(x)$ of increasing degrees, such as $a(x)$ from the list $$ (1+x),(1+x^2),(1+x+x^2),(1+x^3),(1+x+x^3),(1+x^2+x^3),(1+x+x^2+x^3),\ldots $$ and the resulting products are considered to be "random" one can estimate how long on average before one obtains a product trinomial. One need only consider polynomials $a(x)$ with constant coefficient equal to $1.$ There are $2^{k-1}$ such polynomials of degree $k$ since you choose the remaining coefficients to be $0$ or $1,$ so the total number of polynomials $a(x)$ up to degree $k$ is exponential in $k.$ One then estimates what fraction of those products may have only 3 nonzero coefficients and essentially obtain a probability of the form $k 2^{-k}$ for success.
But the reverse question is more useful for my application.