Determine the number of polynomials of degree $5$ with different coefficients in the set $\{1, 2, 3, 4, 5, 6, 7, 8\}$ such that they are divisible by $x^2-x + 1$. Justify your answer.
Problem
Source: OLCOMA Costa Rica National Olympiad, Final Round, 2013 3.5
Tags: algebra, polynomial, divisible