View Single Post
  #9  
Old 04-20-2014, 02:43 AM
RayMan is offline
Guest
 
Join Date: Feb 2004
Location: Chicago, IL USA
Posts: 174
Quote:
Originally Posted by ultrafilter View Post
A polynomial is uniquely determined by its roots and its value at any single point which is not a root. This is pretty easy to show and I'll leave it as an exercise for the reader.
I feel the need to state explicitly what you already know that when you say "...determined by its roots..." you are including the multiplicity of said roots. Yes, I know you know this and that was implied in your statement. I'm making this explicit for the reader doing your exercise.

Consider, for example, a polynomial f(x) with roots at -1, 2, and 3, such that f(0)=6. Then

f(x)=(x+1)(x-2)(x-3)
f(x)=(x+1)^2(x-2)(x-3)

both satisfy these conditions but are manifestly not the same polynomial. The root x=-1 appears with different multiplicity in the two polynomials.