Okay, this is frustrating. I can't seem to prove this elementary result.

Let be distinct, i.e. for . Also, let , i.e. complex nonzero values.

Prove that is a nonzero polynomial in .

I thought about using induction, but I can't see how to make that work. Directly expanding the coefficients is a nightmare, and one which doesn't seem to go anywhere either. I thought maybe there might be some linear algebra theory which would work, but I don't know. Any help would be much appreciated. Thanks!