Problem
NT-B2-M12-P007 Two Roots Give Two Factors
#7
★★★☆☆ Level 3 of 5
Let \(f\in\mathbb Z[x]\), \(f(0)=f(1)=0\). Prove that \(n(n-1)\mid f(n)\) for all integers \(n\).
Use the factor theorem: \(x\) and \(x-1\) divide \(f(x)\).
Since \(f(0)=0\), the polynomial \(x\) divides \(f(x)\) in \(\mathbb Z[x]\). Since \(f(1)=0\), \(x-1\) also divides \(f(x)\). They are coprime, so \(x(x-1)\mid f(x)\) in \(\mathbb Z[x]\). Thus \(f(n)=n(n-1)g(n)\) for some \(g\in\mathbb Z[x]\), proving the divisibility.
Shows the link between polynomial roots and divisibility of values.