Rouche’s Theorem

If the complex-valued functions $f$ and $g$ are holomorphic inside and on some closed contour $K$ , with $|g(z)|<|f(z)|$ on $K$ , then $f$ and $f+g$ have the same number of zeroes inside $K$ , where each zero is counted as many times as its multiplicity.

Example

Consider the polynomial $z^5+3z^3+7$ in the disk $|z|<2$ . Let $g(z)=3z^3+7$ , $f(z)=z^5$ , then

$\begin{aligned} |3z^3+7|&<3(8)+7\\ &=31\\ &<32\\ &=|z^5| \end{aligned}$
for every $|z|=2$ .
Then $f+g$ has the same number of zeroes as $f(z)=z^5$ in the disk $|z|<2$ , which is exactly 5 zeroes.