Problem
NT-B2-M12-P001 Difference of Polynomial Values
#1
★★☆☆☆ Level 2 of 5
Let \(f\in\mathbb Z[x]\). Prove that for any integers \(a,b\), the number \(f(a)-f(b)\) is divisible by \(a-b\).
First prove it for \(x^k\).
For \(k\ge1\), \(a^k-b^k=(a-b)(a^{k-1}+a^{k-2}b+\cdots+b^{k-1})\). A constant term gives zero difference. A sum of such expressions with integer coefficients is also divisible by \(a-b\). Hence the statement holds for every \(f\in\mathbb Z[x]\).
This is the fundamental tool of the module.