  We introduce two new classes of single-valued contractions of polynomial type
defined on a metric space. For the first one, called the class of polynomial
contractions, we establish two fixed point theorems. Namely, we first consider
the case when the mapping is continuous. Next, we weaken the continuity
condition. In particular, we recover Banach's fixed point theorem. The second
class, called the class of almost polynomial contractions, includes the class
of almost contractions introduced by Berinde [Nonlinear Analysis Forum. 9(1)
(2004) 43--53]. A fixed point theorem is established for almost polynomial
contractions. The obtained result generalizes that derived by Berinde in the
above reference. Several examples showing that our generalizations are
significant, are provided.
