Question

The polynomial $(z + i)(z + 2i)(z + 3i) + 10$ has three roots in the complex plane, where $i = \sqrt{-1}.$ What is the area of the triangle formed by these roots?

$6$

$8$

$10$

$12$

$14$

Answer: A

Concepts:complex number polynomial triangle area

Difficulty rating: 1930

Solution:

The sum of the roots is $-6i,$ so the centroid is $-2i.$ Substituting $z = u - 2i,$ $\begin{gathered} (u - i)(u)(u + i) + 10 \\ = u(u^2 + 1) + 10 \\ = u^3 + u + 10. \end{gathered}$

Since $u = -2$ is a root, $u^3 + u + 10$ $= (u + 2)(u^2 - 2u + 5),$ giving roots $u = -2$ and $u = 1 \pm 2i.$

These are the points $(-2, 0),$ $(1, 2),$ $(1, -2).$ The base between $(1, 2)$ and $(1, -2)$ has length $4,$ at horizontal distance $3$ from $(-2, 0),$ so the area is $\dfrac{1}{2}(4)(3) = 6.$ Translation does not change the area.

Thus, the correct answer is A.

2025 AMC 12A Problem 17

Answer: A

Difficulty rating: 1930

Solution:

Problem 17 in Other Years