A basic fact of algebra states that c is a root of a polynomial f(x) if and only if f(x) = (x-c)g(x) for some polynomial g(x). We say that c is a multiple root if f(x) = [(x-c)^2](h(x)) where h(x) is a polynomial.

Question

Show that c is a multiple root of f(x) if and only if c is a root of both f(x) and f'(x)

Answer 1

only if:
f(x) = (x-c)^2 h(x)
f'(x) = 2(x-c) h(x) + (x-c)^2 * h'(x)
= (x-c) * (2h + (x-c)*h^2)

This should get you going. The "if" logic kind of works backwards from here.