Quote:
Originally Posted by ultrafilter
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)(x2)(x3)
f(x)=(x+1)^2(x2)(x3)
both satisfy these conditions but are manifestly not the same polynomial. The root x=1 appears with different multiplicity in the two polynomials.
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