PROBLEM:
Let $f(x)\in F[x]$ with F filed. Suppose there is a root $c$ of f(x) of multiplicity greater than 1 over some extension filed $K$. Prove that $f(x) $ and $f'(x) $ are not relative primes on $F[x] $.
MY ATTEMPT:
Because $c$ is a root, then $f(x) =(x-c)^{2}h(x) $ for some $h(x) \in K[x] $.
On the other hand, it's easy to look that $f'(x) =(x-c) g(x) $ through Leibniz's formula, where $g(x) \in K[x] $
With the above, we can guarantee $gcd(f, f') \neq 1$ on $K[x] $. However, I don't get to prove this property to $F[x] $... Can somebody help me with this?...