  We introduce the concept of piecewise interlacing zeros for studying the
relation of root distribution of two polynomials. The concept is pregnant with
an idea of confirming the real-rootedness of polynomials in a sequence. Roughly
speaking, one constructs a collection of disjoint intervals such that one may
show by induction that consecutive polynomials have interlacing zeros over each
of the intervals. We confirm the real-rootedness of some polynomials satisfying
a recurrence with linear polynomial coefficients. This extends Gross et al.'s
work where one of the polynomial coefficients is a constant.
