Competition problem — Algebra — Maths Olympiad Prep

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Algebra Difficulty 4.5 AIME Find the answer

Suppose $x$ satisfies $x^{3}+x^{2}+x+1=0$ . What are all possible values of $x^{4}+2 x^{3}+2 x^{2}+2 x+1 ?$

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

$x^{4}+2 x^{3}+2 x^{2}+2 x+1=(x+1)\left(x^{3}+x^{2}+x+1\right)=0$ is the only possible solution.

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