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It is a masters level advanced scientific computing course. I ended up figuring it out thanks to a book that had it in it. The first part is actually pretty easy.
Since we know that the integral of P(x)*X^k is equal to zero for k=0..n-1 we know that there is at least 1 zero, look at the case where k=0. We then have the integral of P(x) is equal to zero (thus either P(x) is the zero polynomial or it changes sign at least once).
Now assume P(x) has r < n-1 zeros. And let Q(x) be the interpolating polynomial of P(x) on those zeros. Then P(x)*Q(x) shares zeros with P(x). Thus if they share signs then the integral of P(x)*Q(x)>0 or if they have opposite signs the integral of P(x)*Q(x)<0. But if you expand Q(x) out you get a linear combination of X^k and thus the integral must be zero from the given statements above thus we have a contradiction and P(x) must have more than n-2 zeros.
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