Let . Using Rolle's theorem, prove that is a polynomial of degree n, with distinct roots, all of which lie in the interval (-1,1).
Can anyone help to get me started with this question?
If with satisfying the conditions of Rolle's Theorem we have now we observe that between two roots of there lies a root of . Now observe that has a root at in fact c=0. Now for a roots between (-1,0) and (0,1). Now is indeed an polynomial of order n now use induction to prove that it has distinct roots between (-1,1).