Question 75
A polynomial $f \in \mathbb{Z}[x]$ is called splitty if and only if for every prime $p$, there exist polynomials $g_p, h_p \in \mathbb{Z}[x]$ with $\operatorname{deg} g_p, \operatorname{deg} h_p \lt \operatorname{deg} f$ and all coefficients of $f - g_p h_p$ are divisible by $p$. Compute the sum of all positive integers $n \leq 100$ such that the polynomial $x^4 + 16 x^2 + n$ is splitty.
Proposed by: Pitchayut Saengrungkongka
Proposed by: Pitchayut Saengrungkongka