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What is the difference between the Argument Principal and the Rouche's Theorem.

I am not sure when to use which one when I have a question for example:

How many roots of $z^4 + z^3 + 1=0$ lie inside $C=\{|z|=3/2\}$

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Use whichever one you can get to work for the problem at hand. In the example you gave, you might note that $(3/2)^4 > (3/2)^3+1$, so Rouché should work. If you had $1.3$ instead of $3/2$, it wouldn't be so easy.

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