I am having trouble understanding the proof in my text for the Zeros Theorem. It goes like this:

"Proof: Let P be a polynomial of degree n. By the complete factorization theorem

P(x) = a(x-c1)(x-c2)(x-cn) //the numbers and letter n and i are subscript

Now suppose that c is a zero of P other than c1,c2,.....cn. Then

P(c) = a(c-c1)(c-c2)(c-cn) = 0

Thus, by the Zero-Product Property, one of the factors c-ci, must be 0, so c=ci for some i. It follows that P has exactly the n zeros c1,c2,.....cn."

I was understanding up until they start talking about c-ci must be 0 so c=ci for some i. I'm not sure what they are saying.

Can someone help explain this to me? Thank you in advance!