I'm interested in a special type of polynomial factorization over $\mathbb {Q} $: testing the irreducibility of $f(x)+g(x)$, where $f$ and $g $ are relatively prime and $\text {deg}(f)<\text{deg}(g) $. For ease, let me assume the coefficients are integers.
Surely, there are a bunch of nonexamples; $(2x)+(1+x^2)$, $(x)+(-1-x+x^2)$, and so on. To make the problem meaningful, we should impose some limitations on $f$ and $g$. After a brief googling, I found only two results directly related to the topic above: some Schur-type results http://link.springer.com/article/10.1007/s00605-010-0241-9 and a research paper http://www.worldscientific.com/doi/abs/10.1142/S1793042113500413 .
Hm. How about just monomial $f$, and/or some reducible $g $, with $2\text {deg}(f)\le\text{deg}(g) $? It is the very condition for Schur-type results. In particular, I want to classify(or just find) reducible $g $'s with $\text{deg}(g)=2n$, $f(x)=x^n $ and $g $ having many (at least 11) irreducible factors, making $x^n+g (x) $ reducible. Is it possible?