Where do students learn to solve polynomial equations these days?

The rational root theorem, synthetic division, the remainder theorem, Descartes rule of signs, and similar lower level topics were fairly widely taught in U.S. high school algebra-2 courses before and during the 1980s, but they've slowly been de-emphasized as graphing calculators (allows for numerical equation solving) came onto the scene at the end of the 1980s. Some schools still cover this material as much as ever, and others mostly avoid it.

Regarding more advanced topics such as cubic equations, Vitae's formulas and their uses, etc., these were a standard part of the U.S. undergraduate curriculum when "theory of equations" courses were offered, which was one of the more common upper level math courses a mathematics major would take. These courses were mostly phased out during the mid 1950s to early 1960s as undergraduate abstract algebra courses began taking hold. Older college algebra texts (before 1960s, say) often had some of this material, but I doubt it was actually covered much, if at all, in the standard first year college algebra classes. [Typical college math schedule in U.S. for someone not especially advanced in math, but not necessarily behind either: three 1-semester courses consisting of college algebra (sometimes this was 2 semesters) and trigonometry (sometimes included a bit of spherical trigonometry) and analytic geometry (often included some solid analytic geometry), after which one began studying calculus (often as a 2nd year student, with better students maybe the 2nd semester of their first year).] Since the 1960s, these more advanced topics tend to arise only in abstract algebra courses, typically when Galois theory is covered, but the amount of coverage varies from almost none to a brief treatment, depending on how interested the instructor is in it and how much time is available for what amounts to a diversion on a supplementary topic.

Incidentally, I suspect students are now more aware of many of these topics than they were in the 1990s (but not more than they were in, say, the 1940s) because of the increasing number of (and increased access to information about) mathematics competitions and the rise of the internet, both of which have greatly helped students to become exposed to these topics.