  The Virasoro Lie algebra is a one-dimensional central extension of the Witt
algebra, which can be realized as the Lie algebra of derivations on the algebra
$\cc [t^{\pm}]$ of Laurent polynomials. Using this fact, we define a natural
family of subalgebras of the Virasoro algebra, which we call polynomial
subalgebras. We describe the one-dimensional modules for polynomial
subalgebras, and we use this description to study the corresponding induced
modules for the Virasoro algebra. We show that these induced modules are
frequently simple and generalize a family of recently discovered simple
modules. Additionally, we explore tensor products involving these induced
modules. This allows us to describe new simple modules and also recover results
on recently discovered simple modules for the Virasoro algebra.
