2023 AMC 12B Problems/Problem 14

Problem

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

$\textbf{(A)}\ 5 \qquad\textbf{(B)}\ 6 \qquad\textbf{(C)}\ 8 \qquad\textbf{(D)}\ 7 \qquad\textbf{(E)}\ 4$

Denote three roots as $r_1 < r_2 < r_3$ . Following from Vieta's formula, $r_1r_2r_3 = -6$ .

Case 1: All roots are negative.

We have the following solution: $\left( -3, -2, -1 \right)$ .

Case 2: One root is negative and two roots are positive.

We have the following solutions: $\left( -3, 1, 2 \right)$ , $\left( -2, 1, 3 \right)$ , $\left( -1, 2, 3 \right)$ , $\left( -1, 1, 6 \right)$ .

Putting all cases together, the total number of solutions is $\boxed{\textbf{(A) 5}}$ .

~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)

2023 AMC 12B (Problems • Answer Key • Resources)
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