  We study the twisted Alexander polynomial from the viewpoint of the
SL(2,C)-character variety of nonabelian representations of a knot group. It is
known that if a knot is fibered, then the twisted Alexander polynomials
associated with nonabelian SL(2,C)-representations are all monic. In this
paper, we show that the converse holds for 2-bridge knots. Furthermore we show
that for a 2-bridge knot there exists a curve component in the
SL(2,C)-character variety such that if the knot is not fibered then there are
only finitely many characters in the component for which the associated twisted
Alexander polynomials are monic. We also show that for a 2-bridge knot of genus
g, in the above curve component for all but finitely many characters the
associated twisted Alexander polynomials have degree 4g-2.
