2025 AMC 12A Problems/Problem 17

The polynomial has three roots in the complex plane. What is the area of the triangle formed by these roots?

Solution 1 (Symmetry)

Let be a complex number such that then we can change our polynomial to the following, Notice is a root, and from this we get a quadratic and find the the possible values of

Now we simply subtract from each root to get the roots to be and

Moving back to the coordinate plane our points are and and using shoelace gives us an area of

~mathkiddus