Let F be a field and let f(x) exist in F[x] such that deg f(x) = 3. Prove that f(x) is reducible in F[x] if and only if f(x) has a root in F.
Well I can visualize this proof with graphs, but I am unable write it in a proof. It makes sense that something that is reducible with a deg of 3 would cross the x-axis 2 times, which would make it reducible. But all of this is not formal. How do I begin?
well since deg f(x) is 3 the deg g(x) and deg h(x) can only be either 1 or 3, since exponents are added together when multiplying. So the forward method is making more sense but going the other direction with the roots throws me off. I know that f(x) will cross the axis 3 times but how does that declare that it is reducible?
The degrees actually have to be 1 and 2. Since you have a degree 1, it is of the form for some . Therefore, a root is .
For the other direction, you should recall the factor theorem. If is a polynomial and (i.e. is a root), then for some polynomial . The result is immediate.