  First we construct a free resolution for the Milnor (or Jacobian) algebra
$M(f)$ of a complex projective Chebyshev plane curve $\CC_d:f=0$ of degree $d$.
In particular, this resolution implies that the dimensions of the graded
components $M(f)_k$ are constant for $k \geq 2d-3.$ Then we show that the
Milnor algebra of a nodal plane curve $C$ has such a behaviour if and only if
all the irreducible components of $C$ are rational. For the Chebyshev curves,
all of these components are in addition smooth, hence they are lines or conics
and explicit factorizations are given in this case.
