Question

Let \(n \ge 2\) be an integer and \(p_n (x)\) be the polynomial \[p_n (x) = (x-1) + (x-2) + \cdots + (x-n).\] What is the remainder when \(p_n (x)\) is divided by \(p_{n-1} (x)\)?

  1. \(\dfrac{n}{2}\);

  2. \(\dfrac{n+1}{2}\);

  3. \(\dfrac{n^2+n}{2}\);

  4. \(\dfrac{-n}{2}\).