1. The problem statement, all variables and given/known data Prove that if p(x)=anx^n +an-1x^n-1+..........a0, where a0,.........., "an" ε reals, is a polynomial, then p can have at most n roots. 2. Relevant equations 3. The attempt at a solution C ε R is a root of a polynomial p if p(c)=0. If c is a root of p, then x-c is a factor of p. I'm not sure where to go from here. I think it would probably be the easiest to prove this by proving the contrapositive as being false. Could someone please give me a hint or show me where to go from here? Thank you very much
Sure. Suppose the polynomial has n+1 different roots. c1,c2,...cn+1. Since c1 is a root the polynomial p(x) can be factored (x-c1)*p1(x) where p1 has degree n-1. The other c's must be roots of p1(x) since they aren't roots of (x-c1). Continue in this way until you reach degree 1. Now you have a linear polynomial with two different roots. Possible?