If one needs to use tools from classical invariant theory or elimination theory then some books that come to mind are:
Grace and Young "The Algebra of Invariants", 1903.
Elliott "An Introduction to The Algebra of Quantics", 1913.
Salmon "Lessons Introductory to The Modern Higher Algebra", 1876.
Faa di Bruno "Théorie Des Formes Binaires", 1876.
Faa di Bruno "Théorie Générale de l'Elimination", 1859.
and there are quite a few more.
For Salmon's book, the 4th edition of 1885 might be best. Indeed, as I learned from a paper by Macauley, it has a discussion (on p. 87) of Cayley's very general formula for the multivariate resultant as the determinant of a complex (see the book by Gelfand, Kapranov and Zelevinsky for a modern account and a reprint of Cayley's paper).