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When I was a math undergraduate 30 years ago in India, we were taught what was then called "classical algebra" (as opposed to abstract algebra), and we were taught among other things solving polynomial equations using techniques like Cardan's method, synthetic division, etc.

I haven't kept in touch with math since leaving university, but now when I look at any standard undergraduate course, I don't see any of these things. All I see is abstract algebra.

Do students nowadays not learn how to solve polynomial equations? If not why not?

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.

Students learn linear and quadratic equations in high school algebra. And then, if they have forgotten it, re-learn it in college, in courses called "pre-calculus" or something.

Unless they specialize in mathematics at the college level, they do not learn any more.

Why not? Because we have computers now, so most people do not need to solve polynomial equations by hand any more.

In the United States, solving linear and quadratic equations is a standard part of Algebra 1, which most students take in 8th or 9th grade. Students will return to polynomials and see long division and synthetic division in Algebra 2. This is also when students will learn the remainder theorem, Descartes’ Rule, and how to identify the only possible rational roots for a polynomial.

Most math majors never learn cardano's methods. As for synthetic division, it is part of most high school curricula and linear algebra courses. Cubic and quartic equations aren't part of any curriculum I'm aware of, not because of computers, but because students should learn "more important topics" and because the solutions of these equations are very long so teachers can't really test the students.

Synthetic division is a standard part of the stereotypical "algebra 2" course in the US (~grade 11) and is normally covered including drill problems and examination.

In my experience, cubics and quartics are in the algebra 2 book (I just checked and they are in mine) but in sections with an asterisk. Typically the asterisk sections are not covered because they are considered more difficult diversions and the time, especially for non-superstar students, is better spent solidifying the unasterisked sections. But most teachers will at least say something verbally like" "There are complicated formulas, sort of like the quadratic solution, that can be used to solve general cubics or quartics. We won't cover them because of time, but they are in the book and you should know that methods exist. However, there is no general algebraic solution of 5th or higher order polynomials."

Note that it is standard also to have teaching, drill, and examination on simple (pencil and paper) numerical estimation methods for finding roots to polynomials.

For what it's worth, I think the quadratic really is more fundamental/important in terms of applied problems. Some people say the reason more applied problems use the quadratic is because via circular logic, the students know the equation and this is probably true in some instances. However, we also routinely get people asking questions for fundamental formulas using cubic or higher equations and there really aren't many. Yes, you can construct problems that have enough complication to get to that, but I would not say they are fundamental in the way that the harmonic oscillator is for instance.

For what it's worth, I do think there is a small benefit to covering cubic and quartic methods but would only bother with very high end classes with time to burn. The benefit is from manipulational skill, not from the methods themselves. But being trained, confident in more elaborate manipulations is helpful when you get into physics problems and the ability juggle Taylor series solutions, Bessel functions, Fourier, or change of coordinate systems (without messing up algebraically) is helpful. [Despite what the math and compsci and Maple lovers say, "19th century mathematics is alive and well in physics departments".]

I'm from India and just completed my high school board exams, in our syllabus, we had a chapter called polynomials, in the chapter, we were taught various methods to solve polynomials, those methods were

1. 
   Quadratic formula or referred to as Sri dharacharya formula invented in 930 CE


2. Factorization


3. Competing the square method


4. By using the division algorithm to find the zeros.
   
   

Every indian student has to learn these steps if he's in CBSE board which the major population's choice in our country. This is the syllabus for high school's exam.