multiple roots of a polynomials

Let us say f(x) is a polynomial of degree n.
Also assume f'(x)=0 can be solved easily, say the roots are
x_1,x_2,..,x_{n-1} (counting multiplicity, in some cases). Then
if x_is are not zeros of f(x) (i.e. f(x_i) <>0 for all x_i), can I say
the roots of f(x) are distincts? This theorem or lemma must have a name.
.