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

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.

• Is there a difference between B and D? Aug 14, 2020 at 15:31
• Clearly by seeing u can say it’s -1 Aug 14, 2020 at 15:32
• @NamburuKarthik That's not the only one, though. Aug 14, 2020 at 15:37
• 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. Aug 14, 2020 at 15:43
• Hint: Try to prove by contradiction. Assume $|\alpha|<1$ and use $|x+y+z|\le |x|+|y|+|z|$. Aug 14, 2020 at 15:44

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, $$|\alpha|\ge1$$.