There are several good answers already so I cannot hope to add much. That said, another approach to reach engineers could be by the familiar subject of linear algebra, in particular, solving systems of linear equations as a special case of solving systems of polynomial equations.

Perhaps start with a quick review of linear systems and why we must have either zero, one or an infinite number of solutions. Show the usual pictures in $\mathbb{R}^2$ and $\mathbb{R}^3$ of lines and planes intersecting, including two planes intersecting in a line and a plane and a line intersecting in a point. Use this to lead into discussing solution sets of polynomial equations (with pictures), their dimension, the Hilbert Nullstellensatz and Bezout's Theorem, for instance.