Saturday, May 5, 2018

polynomials - A contest math problem


Let $P(x)$ be a polynomial with integer coefficients of degree $d>0$.




  1. If $\alpha $ and $\beta $ are two integers such that $P(\alpha)=1$ and $P(\beta)=-1$, then prove that $|\beta -\alpha | $ divides $2$.


  2. Prove that the number of distinct integer roots of $(P(x))^2-1$ is at most $d+2$.




First one is very easy. But I cannot understand how to prove the second one. I would appreciate any help.

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