Generalize Exercise 17, showing that f(x) ∈ F[x] has no zero of multiplicity >1 if and only if f(x) and f'(x)
have no common factor in F̅[x] of degree >0.

Data from Exercise 17

Let ƒ(x) ∈ F[x], and let a ∈ F̅ be a zero of f(x) of multiplicity v. Show that v > 1 if and only if α is also a zero of f'(x).