2024 AMC 8 Problems/Problem 22

Problem 22

What is the sum of the cubes of the solutions cubed of $x^5+2x^4+3x^3+3x^2+2x+1=0$ ?

Solution

Factoring $x^5+2x^4+3x^3+3x^2+2x+1$ yields $(x+1)(x^2+1)(x^2+x+1)$ . We can easily find one of the solutions is $x=-1$ . Using the quadratic formula on the rest of the factors yields $-i, i, \frac{-1-i sqrt(3)}{2}$

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2024 AMC 8 Problems/Problem 22

Problem 22

Solution