2024 AMC 8 Problems/Problem 22
Problem 22
What is the sum of the cubes of the solutions cubed of 
?
Solution
Factoring 
yields 
. Denote 
to be solutions of this polynomial. We can easily find one of the solutions is 
. Using the quadratic formula on the rest of the factors yields ![$b=-i, c=i, d=\frac{-1-\sqrt[i]{3}}{2},$](http://latex.artofproblemsolving.com/5/6/6/5663f210800e51c9b431875b1a26d648449bedfb.png)
and finally ![$e=\frac{-1+\sqrt[i]{3}}{2}$](http://latex.artofproblemsolving.com/6/d/7/6d75e008362968cf45d569ff49d172ce42f42b17.png)
. The sum 
is 3, so 3 to the third power is \boxed{27}.
