With a few colleagues, we're trying to design an (intermediate) algebra course (US terminology) where we stress the interplay between algebra and geometry. The algebraic topics we would like to cover are (1) linear equation in two variables, (2) quadratic equations in two variables, (3) polynomials in one variable, (4) rational functions in one variable (though we're not sure we want to introduce functions), (5) radicals.
For (1) and (2) there are obvious geometric counterparts: lines and conic sections.
Question: Are there natural geometric counterparts for (3), (4) and (5)? Are there elementary geometric constructions that naturally lead to these algebraic objects?
Side question: Are there (affordable) textbooks or lecture notes out there which have this kind of approach?