2025 AMC 12A Problem 17

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

Shift by the centroid: let

z = u - 2i

to center the roots at the origin

The polynomial becomes

u^3 + u + 10

= (u + 2)(u^2 - 2u + 5)

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.

Problem 17 in other years