2023 AMC 12B Problem 14

Below is the professionally curated solution for Problem 14 of the 2023 AMC 12B, 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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Concepts:Vieta’s Formulas factor systematic listing

Difficulty rating: 1630

14.

For how many ordered pairs $(a,b)$ of integers does the polynomial $x^3+ax^2+bx+6$ have $3$ distinct integer roots?

$5$

$6$

$8$

$7$

$4$

Solution:

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.

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