Problem
ALG-B1-M04-P010 Values Plus or Minus One
#10
★★★★★ Level 5 of 5
Let \(P(x)\) be a polynomial with integer coefficients. Suppose \(P(2)=1\), \(P(8)=-1\). Prove that no such polynomial exists.
Use the divisibility of \(P(8)-P(2)\) by \(8-2\).
For an integer-coefficient polynomial, \(P(8)-P(2)\) is divisible by \(6\). But the condition gives \(P(8)-P(2)=-2\), which is not divisible by \(6\). Contradiction.
Short, but olympiad-style in its idea.