You have, thus find all real values such as . This only happens for (note the other 4 roots are complex we are only concerned with real). Thus, and are factors. Dividing by we get,Originally Posted bysuedenation

. Thus, the polynomial fractors as

in

To factor in we simply see what are its zero's. This is easy to check because they are only 3 elements in this field. Checking we see that 1 is a zero because, under addition and multiplication modulo 3. Thus, is a factor. Dividing these polynomials we get, (working modulo 3).Originally Posted bysuedenation

Checking the zero's of this polynomial we find that 1 is it zero. Thus, is a factor. Dividing these polynomials we get, (working modulo 3).

, notice this polynomial has no zero and because it has a degree of 3 it is irreducible (this is a theorem about the irreducibility of 2 and 3 degree polynomials). Thus, it factors as,

or another way of writing this,

I do part (c) latter unless you understand and I could stop.