2024 AMC 12A Problems/Problem 15
Contents
Problem
The roots of 
are 
and 
What is the value of ![\[(p^2 + 4)(q^2 + 4)(r^2 + 4)?\]](http://latex.artofproblemsolving.com/9/3/8/938ecf6869b053c8335864493848733a3f242e2e.png)
Solution 1
You can factor 
as (p−2i)(p+2i)(q−2i)(q+2i)(r−2i)(r+2i).
For any polynomial 
, you can create a new polynomial 
, which will have roots that instead have the value subtracted.
Substituting 
and 
into 
for the first polynomial, gives you 
and 
as 
for both equations. Multiplying and together gives you 
.
-ev2028
Solution 2
Let 
. Then
.
We find that 
and 
, so 
.
~eevee9406
See also
| 2024 AMC 12A (Problems • Answer Key • Resources) | |
| Preceded by Problem 14 |
Followed by Problem 16 |
| 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 | |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
