# Thread: Factorization of x in Z_4[x]

1. ## Factorization of x in Z_4[x]

I was hoping someone could give me a hand with this little problem that's been driving me a bit nuts:

Show that there is a factorization x = f(x)g(x) in Z_4[x] (the polynomial ring over the integers mod 4) such that neither f(x) nor g(x) is a constant.

Is there some kind of method or good way to find these factors, or is it just a guess and check situation? I've been searching for these factors for a while with no luck (haven't strayed beyond quadratics yet).

2. Originally Posted by mtdim
Is there some kind of method or good way to find these factors, or is it just a guess and check situation? I've been searching for these factors for a while with no luck (haven't strayed beyond quadratics yet).

Taking into account that

$2\cdot 2=0,\;3\cdot 3=1$

we obtain

$(2x+3)^2=0x^2+0x+1=1$

So, choose

$f(x)=x(2x+3),\;g(x)=2x+3$

Fernando Revilla