2022 AMC 10B Problem 7

Question

For how many values of the constant

k

will the polynomial

x^{2}+kx+36

have two distinct integer roots?

\ 6

\ 8

\ 9

\ 14

\ 16

Different root pairs give different values of k

Integer roots must be factor pairs of

36

Let the roots be

r,s.

Expanding

(x-r)(x-s)

gives

x^2-(r+s)x+rs.

Comparing coefficients with the given polynomial yields

rs=36

and

r+s=-k.

Therefore, we need

r

and

s

distinct such that

rs = 36.

All the possible factor pairs are

\pm\{1,36\},\pm\{2,18\},\pm\{3,12\}

and

\pm\{4,9\}.

Each of these unordered pairs produces a unique value for

k,

so there are

8

possible values for

k.

Thus, B is the correct answer.

Problem 7 in other years