Difference between revisions of "2024 AMC 8 Problems/Problem 22"
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==Solution== | ==Solution== | ||
| − | Factoring <math>x^5+2x^4+3x^3+3x^2+2x+1</math> yields <math>(x+1)(x^2+1)(x^2+x+1)</math>. We can easily find one of the solutions is <math> | + | Factoring <math>x^5+2x^4+3x^3+3x^2+2x+1</math> yields <math>(x+1)(x^2+1)(x^2+x+1)</math>. Denote <math>a, b, c, d, e</math> to be solutions of this polynomial. We can easily find one of the solutions is <math>a=-1</math>. Using the quadratic formula on the rest of the factors yields <math>b=-i, c=i, d=\frac{-1-\sqrt[i]{3}}{2},</math> and finally <math>e=\frac{-1+\sqrt[i]{3}}{2}</math>. The sum <math>a^3+b^3+c^3+d^3+e^3</math> is 3, so 3 to the third power is \boxed{27}. |
Revision as of 13:51, 21 January 2024
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}.
