  Given a polynomial $p$, the degree of its Chebyshev's method $C_p$ is
determined. If $p$ is cubic then the degree of $C_p$ is found to be $4,6$ or
$7$ and we investigate the dynamics of $C_p$ in these cases. If a cubic
polynomial $p$ is unicritical or non-generic then, it is proved that the Julia
set of $C_p$ is connected. The family of all rational maps arising as the
Chebyshev's method applied to a cubic polynomial which is non-unicritical and
generic is parametrized by the multiplier of one of its extraneous fixed
points. Denoting a member of this family with an extraneous fixed point with
multiplier $\lambda $ by $C_\lambda$, we have shown that the Julia set of
$C_\lambda$ is connected whenever $\lambda \in [-1,1]$.

Dynamical Systems