Polynomial roots in C

Question

I have trouble understanding how to prove the following theorem. Let $f$ be a polynomial and let be $c_1,c_2,\dots,c_n$ all different roots of $f$. Then there is an element $c$ in the field so that $$f=c(x-c_1)(x-c_2)\dots(x-c_n)$$ I understand that for the theorem to be true it is needed that the field is algebraically closed, if so there is missing information in the factorization of the polynomial, such as the conjugated roots of every complex root. I hope I have explained myself rather well. If someone could explain why this happens I would be very grateful.