Chebyshev　polynomial　spectral　methods　are　very　accurate,　but　are　plagued　by　the　cost　and　ill-conditioning　of　dense　discretization　matrices.　Modified　schemes,　collectively　known　as　“integration　sparsification”,　have　mollified　these　problems　by　discretizing　the　highest　derivative　as　a　diagonal　matrix.　Here,　we　examine　five　case　studies　where　the　highest　derivative　diagonalization　fails.　Nevertheless,　we　show　that　Galerkin　discretizations　do　yield　banded　matrices　that　retain　most　of　the　advantages　of　“integration　sparsification”.　Symbolic　computer　algebra　greatly　extends　the　reach　of　spectral　methods.　When　spectral　methods　are　implemented　using　exact　rational　arithmetic,　as　is　possible　for　small　truncation　N　in　Maple,　Mathematica　and　their　ilk,　roundoff　error　is　irrelevant,　and　sparsification　failure　is　not　worrisome.　When　the　discretization　contains　a　parameter　L,　symbolic　algebra　spectral　methods　return,　as　answer　to　an　eigenproblem,　not　discrete　numbers　but　rather　a　plane　algebraic　curve　defined　as　the　zero　set　of　a　bivariate　polynomial　P(λ,L);　the　optimal　approximations　to　the　eigenvalues　λj　are　in　the　middle　of　the　straight　portions　of　the　zero　contours　of　P(λ;L)　where　the　isolines　are　parallel　to　the　L　axis.　©　2018　International　Association　for　Mathematics　and　Computers　in　Simulation　(IMACS)