2023 AMC 12B Problem 14

Attempt Problem 14 of the 2023 AMC 12B below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2023 AMC 12B solutions, or check the answer key.

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14.

For how many ordered pairs

(a,b)

of integers does the polynomial

x^3+ax^2+bx+6

have

3

distinct integer roots?

5

6

8

7

4

If the roots are integers

p,q,r,

then

pqr=-6

List the sets of three distinct integers with product

-6;

each set gives one pair

(a,b)

By Vieta, the three distinct integer roots multiply to

-6.

The sets of three distinct integers with product

-6

are

\{1,2,-3\},

\{1,-2,3\},

\{-1,2,3\},

\{-1,-2,-3\},

and

\{1,-1,6\}.

Each set determines

a=-(p+q+r)

and

b=pq+qr+rp,

and all five give different pairs, so there are

5

ordered pairs

(a,b).

Thus, the correct answer is A.

Problem 14 in other years