Proving a Formula: Maximum Root Count for Polynomials

[chocolatelover]

Homework Statement

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.

Homework Equations

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

 

[Dick]

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?

 

[chocolatelover]

Thank you very much

Would it be somthing like this?

(p1x)^(n-1)(x-c2)(x-c3)^(n)

Thank you

 

[Dick]

No, that's not clear at all. Start by proving if n=1 then the polynomial can't have 2 roots. Ok?

 

[chocolatelover]

Thank you very much

Regards