  We study the Chebyshev-Halley family of root finding algorithms from the
point of view of holomorphic dynamics. Numerical experiments show that the
speed of convergence to the roots may be slower when the basins of attraction
are not simply connected. In this paper we provide a criterion which guarantees
the simple connectivity of the basins of attraction of the roots. We use the
criterion for the Chebyshev-Halley methods applied to the degree $n$
polynomials $z^n+c$, obtaining a characterization of the parameters for which
all Fatou components are simply connected and, therefore, the Julia set is
connected. We also study how increasing $n$ affects the dynamics.
