Although I am not adding to anything said above, let me summarize by saying that two polynomials of the same degree n with n+1 points in common are the same. If you want these points to be the roots, you need one more and if 0 is a root, having the same value at 0 won't give a new point. And if one or more is a double root, you won't have n+1 points, although in that case you have a root of the derivatives and beyond that it gets a bit complicated. But if they have the same roots with the same multiplicities, then you do need one more point (so that must be a nonroot). And of course, you must include all complex roots.
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