2024 AMC 12A Problems/Problem 15

The roots of x^3 + 2x^2 − x + 3 are p, q, and r. What is the value of (p^2 + 4)(q^2 + 4)(r^2+4)?

Solution

You can factor (p^2 + 4)(q^2 + 4)(r^2 + 4) as (p − 2i)(p + 2i)(q − 2i)(q + 2i)(r − 2i)(r + 2i).

For any polynomial f(x), you can create a new polynomial f(x+2), which will have roots that instead have the value subtracted.

Substituting x-2 and x+2 into x for the first polynomial, gives you 10i-5 and -10i-5 as c for both equations. Multiplying 10i-5 and -10i-5 together gives you 125

-ev2028