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If $\alpha$ is a root of $Z^5+Z^3+Z+3=0$, then find $\alpha$.

ORIGINAL QUESTION

If $\alpha$ is a root of $Z^5+Z^3+Z+3=0$, then

A) $|\alpha| \geq 1$

B) $|\alpha|<1$

C) $\alpha$ lies on or outside the circle $|z|=\frac{1}2$

D) $\alpha$ lies inside the unit circle $|z|=1$

My Attempt: I tried using Rouche’s theorem, but I am unable to get my desired answer.

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  • $\begingroup$ Is there a difference between B and D? $\endgroup$
    – Arthur
    Aug 14 '20 at 15:31
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    $\begingroup$ Clearly by seeing u can say it’s -1 $\endgroup$ Aug 14 '20 at 15:32
  • $\begingroup$ @NamburuKarthik That's not the only one, though. $\endgroup$
    – Arthur
    Aug 14 '20 at 15:37
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    $\begingroup$ First $B,D$ cannot be true for all roots because they are not true for $-1$. Next if $A$ is true then $C$ must be true, no matter what the number is. And there is only one correct answer. $\endgroup$
    – cr001
    Aug 14 '20 at 15:43
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    $\begingroup$ Hint: Try to prove by contradiction. Assume $|\alpha|<1$ and use $|x+y+z|\le |x|+|y|+|z|$. $\endgroup$
    – SarGe
    Aug 14 '20 at 15:44
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Assume that $|\alpha|\lt 1$. We have, $$\Big(|\alpha^5+\alpha^3+\alpha|=3\Big)\le|\alpha|^5+|\alpha|^3+|\alpha|<3$$ which is a contraction. Hence, $|\alpha|\ge1$.

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  • $\begingroup$ Thank you @SarGe $\endgroup$
    – Anonymous
    Aug 14 '20 at 16:54
  • $\begingroup$ You're welcome! :-) $\endgroup$
    – SarGe
    Aug 14 '20 at 17:06

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