Applying Argument Principle to find zeroes in a quadrant when there is a real root

Question

I was trying to find the number of roots of the polynomial $p(x)=z^9+2z^5-2z^4+z+3$ that lie in the second quadrant. I know that $p(-1)<0$ and $p(0)>0$, so there is some real root on the negative real axis between $-1$ and $0$. This means that I cannot use either Rouche's Theorem or the Argument Principle to find number of roots in the second quadrant by checking the strict inequality/change in argument around the boundary of the quarter disc in the second quadrant, because there is a zero on $\delta D$. I tried applying argument principle on the quarter disc where instead of the negative real axis, I took it along $re^{i(\pi - \varepsilon)}$, but still got nowhere. Any help on how to go about this would be appreciated!

Answer 1

We can use a few other tools quickly to form a sneaky strategy:

• It's not hard to check by hand that $p(z)$ has no zeros on the imaginary axis (we can determine the zeros of its real and imaginary parts there).
• Since $p(z)$ has real coefficients, its complex zeros come in conjugate pairs.
• $p(z)$ has exactly one negative zero, as you've found.
• Therefore the number of zeros in the left half-plane will be $2N+1$, where $N$ is the number of zeros in the second quadrant.