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?

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

Solution

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)

Video Solution

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

See Also

 2023 AMC 12B (Problems • Answer Key • Resources) Preceded byProblem 13 Followed byProblem 15 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 All AMC 12 Problems and Solutions

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