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The polynomial P is even if it is even for a,b

Source: Baltic Way 1998

January 11, 2011
algebrapolynomialalgebra proposed

Problem Statement

Let PP be a polynomial of degree 66 and let a,ba,b be real numbers such that 0<a<b0<a<b. Suppose that P(a)=P(−a),P(b)=P(−b),P′(0)=0P(a)=P(-a),P(b)=P(-b),P'(0)=0. Prove that P(x)=P(−x)P(x)=P(-x) for all real xx.
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