What is the remainder when $p_n (x)$ is divided by $p_{n-1} (x)$?

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

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

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