Exercise
"5.5 Determine which of the following polynomials are
irreducible over Z_2[x]: x ^ 5 + x ^ 4 + 1,
x ^ 5 + x ^ 3 + 1, x ^ 5 + x ^ 4 + x ^ 2 + 1."
-Douglas R. Stinson-
See "Cryptography: Theory and Practice" by Douglas R.
Stinson pages 200-201.

x ^ 5 + x ^ 4 + 1 = (x ^ 2 + x + 1) * (x ^ 3 + x + 1) mod 2
hence, it is not irreducible over Z_2[x]

x ^ 5 + x ^ 3 + 1 is irreducible over Z_2[x]

x ^ 5 + x ^ 4 + x ^ 2 + 1 = (x + 1) * (x ^ 4 + x + 1) mod 2
hence, it is not irreducible over Z_2[x]