2022 AMC 12B 第 4 题

If the roots are $p$ and $q,$ then $pq = 36$ and $k = -(p+q)$

Count the ordered-independent factor pairs of $36$ with two distinct integers, including negatives

If the roots are integers $p$ and $q,$ then $pq = 36$ and $k = -(p+q).$ Distinct roots must have the same sign, so we list factor pairs of $36$ with $p \ne q.$

The positive pairs are $(1,36), (2,18), (3,12), (4,9),$ and the negative pairs are $(-1,-36),$ $(-2,-18),$ $(-3,-12),$ $(-4,-9).$ The pair $(6,6)$ is excluded since the roots must be distinct.

Each of these $8$ pairs gives a different value of $k.$

Thus, the correct answer is B.