Your answers should be polynomials over F2 with degrees at most four. (Can you explain in here why at most degree four)
Note: Use the Eucliden algorithm for polynomials. (Can you do this algorithm step by step to understand)
All the following calculation are in modulo x^5+x^2+1 and we note 1=-1 in F2.
By Euclidean algorithm, we have
0=x^5+x^2+1=(x^2-1)(x^3+x)+(x^2+x+1) => (x^2-1)(x^3+x)=x^2+x+1
x^3+x = ...
Irreducible Polynomials are investigated.