Helping students gain understanding and self-confidence in algebra.
jwroblewski44 wrote:Proof: Let P be a polynomial of degree n. By the complete factorization theorem, P(x) = a(x - c1)(x - c2) ... (x - cn). 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?
jwroblewski44 wrote:So when they say 'ci' they aren't using i to mean sqrt(-1)? The use of 'i' has really confused me.
jwroblewski44 wrote:EDIT: A question I forgot to mention in my OP, how is there a zero, c, that is not c1,c2....cn?
jwroblewski44 wrote:I still can't see how this proof shows us how every polynomial has exactly as many factors as the degree of the polynomial?