Suppose the root u appears more than once in p(x). In other words, the complete factorization of p(x) in the field F includes (x-u)m, where the multiplicity m is at least 2.
Let q be the formal derivative of p, then take the gcd of q and p to find any repeated factors. Since x-u appears in both q and p, the gcd is nontrivial, and p is not irreducible after all. This procedure always works when the characteristic of K is 0.
In summary, a field with characteristic 0 is perfect. All its extensions are separable. This includes the rationals, and various algebraic extensions of the rationals, but it also includes fields like Q(x,y,z), the rationals adjoin three indeterminants. Call this field K and note that it has characteristic 0. Thus every extension of K is separable, end of story.