2022 AMC 12B Problem 4

For how many values of the constant

k

will the polynomial

x^2 + kx + 36

have two distinct integer roots?

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Solution

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.