How many solutions lie in the left half-plane?
$$f(z) = z^3+2z^2-z-2+e^z=0$$
My work so far:
Factoring the polynomial, moving the exponential term over to the RHS, and taking the modulus of both sides gives me:
$$|(z+1)(z+2)(z-1)| = |e^{-x}| = \frac{1}{e^x} < 1$$
The obvious numbers that fulfill the above inequality are the roots -1,-2,+1, which makes the LHS = 0.
Among these numbers, -1 and -2 are from the left half-plane, so I have two solutions.
...do I have more solutions? How could I apply Rouche's Theorem here?
Any ideas are welcome.
Thanks,