Create the polynomial p(x)=x5+2x4+1 over Z3. Verify that it does not have any linear factors by evaluating p(x) with each element of Z3, and then check that p(x) is irreducible.
Create a finite field of order 35 with the FiniteField() command, but include the modulus keyword set to the polynomial p(x) to override the default choice.
Recreate p(x) as a polynomial over this field. Check each of the 35=243 elements of the field to see if they are roots of the polynomial and list all of the elements which are roots. Finally, request that Sage give a factorization of p(x) over the field, and comment on the relationship between your list of roots and your factorization.