Let F be a field and let f (x), g(x) ∈ F[x]. Show that N = {r(x)f(x) + s(x)g(x) | r(x), s(x) ∈ F[x]} is an ideal of F[x]. Show that if f(x) and g(x) have different degrees and N ≠ F[x], then f(x) and g(x) cannot both be irreducible over F.