TITLE: Determining the Number of Zeros of a (Holomorphic) Polynomial $f:\mathbb{C}\to\mathbb{C} $ in each Quadrant.
QUESTION [5 upvotes]: Suppose $f(z)=z^4+2z^3+3z^2-z+2$. I would like to be able to determine the number of zeros (without using a CAS) $f$ has in each quadrant.
I recently learned about the Argument Principle and Rouche's Theorem. I can use Rouche's Theorem to estimate the number of zeros of $f$ on a given disc restricted to one of the quadrants, but how do I find which disc to use?
The other idea I have in mind is to create arbitrary discs in the plane and to apply the argument principle to see how many zeros $f$ has in each disc, but this doesn't seem like an efficient way to go for the same reason as why I wouldn't use Rouche's Theorem.
Is there an efficient way to determine how many zeros are in each quadrant? I am looking for a technique general enough to be applied to any polynomial.

REPLY [5 votes]: Let $\Gamma _1: z=x$ $(0\le x\le R)$, $C: z=Re^{i\theta } $ $( 0\le \theta \le \pi/2)$
 and $\Gamma _2: z=i(R-y)$ $(
 0\le y\le R)$ as in Fig.1.
We note 
several facts:
(1) For $z=x$ real, $f(x)=x^4+2x^3+3x^2-x+2>0 $ $(\, -\infty< x<\infty)$,
(2) For sufficiently large $z$, $\arg f(z)=\arg z^4(1+o(1/z))=\arg z^4+\arg (1+o(1/z))
=4\arg z+\varepsilon $,
(3) For $z=iy$, where $y$ is real positive, $\Im f(iy)=-2y^3-y<0$.
From these facts, $f(\Gamma _1), f(C)$ and $f(\Gamma _2)$ are like as in Fig.2 if $R$ 
is suffiently large. They go around the origin once,  which means that the number of 
zeros of $f(z)$ inside the contour $\Gamma _1 \cup C 
\cup \Gamma _2$ is $1$ by the argument principle.
Same arguments can be applied for quadrant II and the number of zeros of $f(z)$ inside 
quadrant II is $1$. 
If $\alpha $ is a root of $f(z)=0$, then $\overline{\alpha}$ is also 
a root of it. Thus the number of zeros of $f(z)$ is $1$ in each quadrant.