How can we show that the polynomial
$a_nx^n + a_{n-1}x^{n-1} + a_3x^3 + x^2 + x + 1 = 0$, where $a_i\in \Bbb R$, $i=3,...,n$
has an imaginary root?
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This is a special case of the fundamental theorem of algebra. So the most simple way to prove your statement is to refer to that theorem. In case you're supposed to solve the problem without the mentioned theorem, you may see the most known proofs here: https://en.wikipedia.org/wiki/Fundamental_theorem_of_algebra#Proofs – Andrei Rykhalski Dec 08 '14 at 02:21
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I've spent a couple of minutes trying to think how to answer this question. I don't see how to prove that it's not the case that all of its roots are real. My only thought is the fact that by Descartes' rule of signs, not all of its roots are positive. – Tanner Swett Dec 08 '14 at 02:23
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Note that the term "imaginary number" typically refers to a pure imaginary number, i.e., $ai$ where $a \in \mathbb R$ and $i^2 = -1$. For that interpretation of this problem, it is false; take, for instance, $a_i = 1$ for all $i$, and $n$ to be odd. – Dustan Levenstein Dec 08 '14 at 02:25
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Maybe play around with the roots of unity? – Passing By Dec 08 '14 at 03:11