The application of Rouche's theorem

Question

As shown in the picture, I am wondering how to decide which power to substract? I don't understand it. Thanks.

Answer 1

In first part, Take $f(z) = z^4+3z^3+6$ and $g(z)=3z^3$. Then, $$|f(z)-g(z)|< |f(z)|+|g(z)|$$ evidently on the contour $ |z|=2$ as shown. Hence Rouché's theorem can be applied given us the number of zeroes for $f$ and $g$ inside the contour given as 3.

Similarly, in the second part, Take $f(z)=z^4-2z^3+9z^2+z-1$ and $g(z)=9z^2$.$\implies |f(z)-g(z)|< f(z)+g(z)$ on the contour $|z|=2$ thus giving us the same number of zeroes for $f$ and $g$ inside the contour by Rouché's theorem.

Answer 2

You can just reduce the problem to the unit circle $|w|=1$ by computing \begin{align} f_1(2w)&=16w^4+24w^3+6,\\ f_2(2w)&=16w^4−16w^3+36w^2+2w−1. \end{align} In both cases it is easy to confirm that the largest coefficient is larger than the sum of the absolute values of the other coefficients, even if barely, and thus the dominant one.