  The Gauss--Lucas and B\^{o}cher--Grace--Marden theorems are classical results
in the geometry of polynomials. Proofs of the these results are available in
the literature, but the approaches are seemingly different. In this work, we
show that these theorems can be proven in a unified theoretical framework
utilizing matrix analysis (in particular, using the field of values and the
differentiator of a matrix). In addition, we provide a useful variant of a
well-known result due to Siebeck.
