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 \(p_{n-1} (x)\)?

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

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

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

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

The terms in \(p_n (x)\) follow a sequence - can we sum this?

What value of \(x\) makes \(p_{n-1}(x)\) zero?