In the context of a family of scalar-tensor theories with a dynamical
$\Lambda$, that is a binomial on the scalar field, the cosmological
equations are considered.  A general barotropic state equation
$p=(\gamma-1)\rho$, for a perfect fluid is used for the matter content
of the Universe.  Some Friedmann-Robertson-Walker exact solutions are
found; they have a scale factor which shows exponential or power law
dependence on time. For some models the singularity can be avoided.
Cosmological parameters such as $\Omega_m$, $\Omega_{\Lambda}$, $q_0$ 
and $t_0$ are obtained and compared with observational data.