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I need to prove that the equation $x^4 + x^2 + 1 = 0$ has no solution in $\mathbb{R}$.

How would I go about proving this?

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  • $\begingroup$ Have you tried factoring the left-hand side? $\endgroup$ – Nick Dec 11 '14 at 21:07
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$x^2$ and $x^4$ are not negative, so $$1+x^2+x^4\ge1$$ for any real number $x$.

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  • $\begingroup$ This is better than my answer. On the bright side, at least my answer will work. $\endgroup$ – Matt Samuel Dec 11 '14 at 21:03
  • $\begingroup$ This is the type of proof I was looking for! It seems obvious after reading your answer, but how would you come up with that in the first place? $\endgroup$ – GWAO Dec 11 '14 at 21:07
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    $\begingroup$ Hard to resist upvoting this answer, plus one! $\endgroup$ – Robert Lewis Dec 11 '14 at 21:10
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    $\begingroup$ @GWAO Just think real numbers with even exponents as nonnegative numbers. $\endgroup$ – ajotatxe Dec 11 '14 at 21:40
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I would solve the equation. Take $y=x^2$. Then the equation is $y^2+y+1=0$. This is quadratic and easy to solve.

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  • $\begingroup$ I tried that! It seemed a little too simple to be true haha $\endgroup$ – GWAO Dec 11 '14 at 21:07
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    $\begingroup$ @GWAO the lesson is that sometimes math is as simple as it looks! $\endgroup$ – Matt Samuel Dec 11 '14 at 21:08
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Easy! Since $x^4 \ge 0$ and $x^2 \ge 0$ for all $x \in \Bbb R$, we have $x^4 + x^2 \ge 0$, whence $x^4 + x^2 + 1 > 0$, $\forall x \in \Bbb R$. No zeroes in $\Bbb R$, no solution! QED!!!

Of course, if we allow $x \in \Bbb C$, we have a different story altogether, which is told in a different place.

Hope this helps. Cheers!

And as ever,

Fiat Lux!!!

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    $\begingroup$ Do you think you could use a few more exclamation points in there? ;) $\endgroup$ – augurar Dec 11 '14 at 21:56
  • $\begingroup$ No such thing as too many exclamations. Helps get your point across. (!!!) $\endgroup$ – Dasherman Dec 11 '14 at 22:02
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    $\begingroup$ @Dasherman: I couldn't agree more !!!!!!!!!!!!!!!!!!!!! $\endgroup$ – Robert Lewis Dec 11 '14 at 22:28
  • $\begingroup$ @augurar: try 'em, you'll like 'em . . . Hell! You'll Love 'em !!!!!!!!!!!!!!!!!!!!! $\endgroup$ – Robert Lewis Dec 11 '14 at 22:29
  • $\begingroup$ And a Jovial Yule Season to All and Sundry !!!!!!!!!!!!!!!!!!!!! Ho ho ho!!!!!!!!!!!!!!!!!!!!! $\endgroup$ – Robert Lewis Dec 11 '14 at 22:29

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