Problem
NT-B2-M12-P018 Leading Coefficient of an Integer-Valued Polynomial
#18
★★★★★ Level 5 of 5
Let \(P(x)\) be a degree \(d\) polynomial with rational coefficients that takes integer values at every integer \(x\). Prove that \(d!\) times the leading coefficient of \(P\) is an integer.
Consider the \(d\)-th finite difference \(\Delta^d P(0)\).
The operator \(\Delta P(x)=P(x+1)-P(x)\) sends integer values to integer values. Therefore \(\Delta^dP(0)\) is an integer. If the leading coefficient of \(P\) is \(c\), then each application of \(\Delta\) lowers the degree by \(1\) and multiplies the leading coefficient by the current degree. After \(d\) applications, we get the constant polynomial \(d!c\). Hence \(d!c=\Delta^dP(0)\in\mathbb Z\).
This is a strong fact, but the proof is short and very useful.