For p(x)∈C[x] such that ∫10p(x)xkdx=0 for 0≤k≤n−1, show that p(λ)=0⇒λ∈[0,1]

manpreet Best Answer 3 years ago

For a complex polynomial p(x)∈C[x]p(x)∈C[x] of degree nn such that ∫10p(x)xkdx=0∫01p(x)xkdx=0 for 1≤k≤n−11≤k≤n−1, show that p(λ)=0p(λ)=0 means λ∈[0,1]λ∈[0,1].

I haven't come by any theoretical direction so far. Since it is a question given in an Introduction to Operator Theory and Hilbert Spaces, I think that approaching Complex Analysis only will be very tedious. The problem is that I can't really tell what of the the course themes it is that I should use. I was thinking Min-Max theorem, Spectral Theory and things in that area of the course syllabus. I still can't see how it can be done; the question seems irrelevant. I would appreciate it if you could give any hint or guide me a little.