if the polynomial has at most n roots 0, but the polynomial is given to be 0 for EVERY x, what does this tell us about the ?

to make it sharper, suppose is not the 0-polynomial. then we can find some x for which is NOT 0, contradicting our choice of coefficients.

alternatively, note that R[x] is a subspace of the continuous function space (polynomials are continuous functions). the set is a basis for R[x], and we can extend any basis of a subspace to a basis for the enitre space. then must be linearly independent, being a subset of a basis.