  In this paper, the problem of bounding the number of reducible curves in a
pencil of algebraic plane curves is addressed. Unlike most of the previous
related works, each reducible curve of the pencil is here counted with its
appropriate multiplicity. It is proved that this number of reducible curves,
counted with multiplicity, is bounded by d^2-1 where d is the degree of the
pencil. Then, a sharper bound is given by taking into account the Newton's
polygon of the pencil.
