Rouche's theorem in the right half plane

Question

I need to determine the number of zeros in the right half plane Re z>0 of the polynomial: $$ f(z)=z^3-z+1 $$ My attempt to solve the problem:

I'm using Rouché's theorem and consider $$ g(z)=z^3+1 ~~ and ~~ p(z)= -z $$ and it can be seen that $|g(z)|>|p(z)|$ on the imaginary axis and for $|z|$ large.

My conclusion is that the function $f(z)$ has $3$ zeros in the right half plane. Is this correct?

(A solution from the book showed that it has $2$ zeros in $Re\, z>0$ but I'm not sure if it's a typo)

Answer 1

You correctly derived that $f$ and $g$ have the same number of zeros in the right half-plane.

Only the final conclusion is wrong: $g(z) = z^3+1$ has two zeros in the right half-plane (the third zero is $z=-1$).