In 2013, Derksen showed that the 3x3 determinant polynomial can be written as a sum of five products of linear forms. This begs the question: Can one get by with fewer than five summands? Zach Teitler and I have shown that five summands are in fact necessary. Our argument relies on an analysis of the linear subspaces contained in a special hypersurface. Motivated by this application, I will discuss recent work concerning linear subspaces contained in the vanishing loci of certain special sets of polynomials, such as determinantal varieties and toric varieties.