MathDB
polynomial

Source: flanders '96

September 27, 2005
algebrapolynomial

Problem Statement

Consider a real poylnomial p(x)=anxn+...+a1x+a0p(x)=a_nx^n+...+a_1x+a_0. (a) If deg⁡(p(x))>2\deg(p(x))>2 prove that deg⁡(p(x))=2+deg(p(x+1)+p(x−1)−2p(x))\deg(p(x)) = 2 + deg(p(x+1)+p(x-1)-2p(x)). (b) Let p(x)p(x) a polynomial for which there are real constants r,sr,s so that for all real xx we have p(x+1)+p(x−1)−rp(x)−s=0 p(x+1)+p(x-1)-rp(x)-s=0 Prove deg⁡(p(x))≤2\deg(p(x))\le 2. (c) Show, in (b) that s=0s=0 implies a2=0a_2=0.
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