Rekna

If anyone can give me some hints beyond the ones I already have i'd be greatful.

Notation:
int--> stands for integreal

Here is the problem

Let P be a real polynomial of degree n such that:

int[a,b] P(x)*X^k dx = 0 where k=0,...,n-1.

a) Show that P has n real zeros (simple) in (a,b).

Hint: Show that p must have at least one zero in [a,b], and that assumption tha tp has less than n zeros yields a contradiction; (consider int[a,b] P(x)Qn-1(x) dx, where Qn-1(x)=(x-x1)...(x-xn-1) and Xi i=1,..,n-1 are the zeros of P in (a,b).

b) Show that the n-point interpolating rule on [a,b] based on zeros of p has degree 2n-1.

Hint: Consider the quotient and remainder polynomials whena given polynomial of degree 2n-1 is divided by p.


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Showing a zero exists in P is easy. Take the case where k=0 then we have the int[a,b] P(x) = 0. This means that P(x) must change signs at least once and hence has a zero.

But this is as far as I can get. I think my problem is I have very little background in this subject matter especially when it gets into the orthogonal inner products occuring when we multiply by X^k. Any suggestions of would be great.