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For something like $$x^7 + 2x^3 + 3x - 7$$

Would I apply Rolle's theorem ?

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    $\begingroup$ What does this question have to do with group theory? $\endgroup$ – 5xum Mar 3 '16 at 10:50
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    $\begingroup$ the example function you provided is monotonic increasing... $\endgroup$ – Harry Mar 3 '16 at 10:55
  • $\begingroup$ Yes, Rolle's theorem helps a lot. $\endgroup$ – bartgol Mar 3 '16 at 21:26
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Hint: the intermediate value theorem tells you that there is a real root for your polynomial. Then if you differentiate you get:

$$7x^6 + 6x^2 + 3$$

which is always positive. You can easily conclude form that.

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Since $x^7, x^3$ and $x$ are all strictly increasing, their sum is strictly increasing. The "$-7$" is just a translation. It follows the function is strictly increasing, therefore injective. In particular, there can be only one zero.

Observing that $$\lim_{x\to -\infty} f(x) = -\infty$$ and $$\lim_{x\to+ \infty} f(x) = +\infty$$ and that $f$ is continuous, there is at least one so you're done.

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