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This is the question: How many real solutions is there to the following equation?

$ x+x^2 +...+ x^{2012} = 0$

The answer is the 2 solutions: $x=0$ and $x=-1$

I solved the question by just factorizing the expression. But I was wondering if these types of problems belong to some category in which you should solve with a specific technique (in case the problem is much harder). If so, what are some good sites to read about it?

5TableLegs
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    How familiar are you with complex numbers? To address your question, there is no simple answer in general, but this particular polynomial is special. There are plenty of special polynomials, belonging to plenty of different fields, if varying difficulty. Your question is basically impossible to give a good answer to. – Arthur Nov 06 '22 at 15:19
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    In general, it is not easy to determine the number of real roots of a polynomial. The sturm chains allow to determine it for an arbitary polynomial , but usually it can only be done with a computer in a reasonable amount of time. – Peter Nov 06 '22 at 15:26
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    I know what complex numbers are and how to calculate them, that's all. The question is taken from an entry-level IMO-team recruitment test. I don't know if that gives you information about the best solving method. – 5TableLegs Nov 06 '22 at 15:27
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    In a high school age competitive setting, this is just a polynomial you are expected to recognise and know a bunch about. For instance what happens if you multiply by $1-x$. – Arthur Nov 06 '22 at 15:31
  • Or you can use geometric series formula (It holds for $x\neq1$ and note that $x=1$ is not a root of the equatio so there is not a problem). – Etemon Nov 06 '22 at 16:27

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