While I have obtained a PhD in maths, I work as teacher for maths and physics at a secondary school by now and am certainly no proper research mathematician. Still, I am seriously interested in the geometry of polynomials and use Gauss-Lucas-matrices to introduce basic concepts of higher mathematics to interested pupils. Discussing Gauss-Lucas-matrices allows to treat diagonalizability, fourier series and quadratic number fields among many other things in relation to (geometry of) polynomials.