  This work studies the zeros of slice functions over the algebra of dual
quaternions and it comprises applications to the problem of factorizing motion
polynomials. The class of slice functions over an alternative $*$-algebra $A$
was defined by Ghiloni and Perotti in 2011, extending the class of slice
regular functions introduced by Gentili and Struppa in 2006. Both classes
strictly include the polynomials over $A$. We focus on the case when $A$ is the
algebra of dual quaternions $\mathbb{DH}$. The specific properties of this
algebra allow a full characterization of the zero sets, which is not available
over general alternative $*$-algebras. This characterization sheds some light
on the study of motion polynomials over $\mathbb{DH}$, introduced by Heged\"us,
Schicho, and Schr\"ocker in 2013 for their relevance in mechanism science.
