Let f(x) and g(x) be nonzero polynomials in R[x] and assume that the leading coefficient of one of them is a unit. Show that f(x)g(x) doesn't equal 0 and that deg[f(x)g(x)] = deg(f(x)) + deg(g(x))
Without the loss of generality, we suppose the leading coefficient of f(x) is a unit. We can suppose f(x)=anx^n+...+a1x+a0, g(x)=bmx^m+...+b1x+b0, where ...
A polynomial proof is provided.