Remark

The converse is also valid: If f R[X] and h is an irreducible (nonconstant) factor of gcd(f,Df) 1, then, h is a multiple factor of f.

For, writing f = gh, we have gD(h) = Df - D(g)h = 0 mod h.

If Dh = 0, then h is a nonconstant polynomial in X^p, whence the pth power of a nonconstant polynomial, and so f has a multiple factor.

Suppose now Dh 0. Since h is irreducible and Dh has degree less than deg(h), we find gcd(D(h),h) = 1. But then gD(h) = 0 mod h implies g = 0 mod h. This shows that h, being a divisor of both g and h, is a multiple factor of gh = f.