  An explicit expression for the cofactor related to an irreducible invariant
algebraic curve of a polynomial dynamical system in the plane is derived. A
sufficient condition for a polynomial dynamical system in the plane to have a
finite number of irreducible invariant algebraic curves is obtained. All these
results are applied to Li\'enard dynamical systems $x_t=y$, $y_t=-f(x)y-g(x)$
with $\text{deg}\, f<\text{deg}\,g<2\,\text{deg}\,f+1$. The general structure
of their irreducible invariant algebraic curves and cofactors is found. It is
shown that Li\'enard dynamical systems with $\text{deg}\, f<\text{deg}\,
g<2\,\text{deg}\, f+1$ can have at most two distinct irreducible invariant
algebraic curves simultaneously and consequently are not integrable with a
rational first integral.
