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I am guessing solving general cubic and quartic equations should be taught in a course somewhere between precalculus and Galois theory, though personally I do not recall learning this topic ever in any course.

When should we teach students (mainly ones majoring in mathematics) how to solve general cubic and quartic equations? Or solving general cubic and quartic equations is so unimportant that it does not even deserve to be taught to even undergraduates majoring in mathematics?

Edit: I understand that it is impractical to solve such equations by hand. On the other hand, are those algorithms at least of some theoretical importance that an undergraduate student majoring in mathematics should know the algorithm before his graduation?

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    $\begingroup$ My answer. Numerical solution in calculus. Solution in radicals should not be taught before Galois theory. And even in Galois theory maybe you do only the proof that solution in radicals is possible for cubic and quartic, but do not waste time carrying out all the cases. $\endgroup$ – Gerald Edgar Dec 16 '18 at 21:15
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    $\begingroup$ I don't remember learning them also. Jacobson put them after groups, rings and modules but before polynomials. Here's the ToC: 1, 2 and 3. $\endgroup$ – Paracosmiste Dec 16 '18 at 21:26
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    $\begingroup$ The only such equations students are likely to meet are those cooked up to give an easy solution (e.g. simple trig functions in the case of a cubic). Otherwise, this is just a pointless exercise in pushing buttons on a calculator. Try solving a "real world" quartic like $5.683x^4 - 1.273x^3 + 7.495x^2 - 2.437x + 9.236 = 0$, see how long it takes you. The only practical thing you will learn is that life is too short to do this sort of stuff by hand. $\endgroup$ – alephzero Dec 17 '18 at 0:30
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    $\begingroup$ If you really want to teach how to solve polynomials, wait till the linear algebra course and show them how to solve high-order polynomials (high = 300 or 400, not 3 or 4) by constructing a matrix with the given polynomial as its characteristic equation, and then solving the eigenproblem numerically. Wilkinson's (in)famous example of an ill-conditioned 20th-order polynomlial is instructive, if solved that way. $\endgroup$ – alephzero Dec 17 '18 at 0:40
  • $\begingroup$ They make good honors math topics for highschool. My wife worked them out in her advanced-level math in Hong Kong. In my current teaching, I put it near by Galois theory where it can be appreciated. $\endgroup$ – James S. Cook Dec 17 '18 at 17:10
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The place I’ve seen this is usually in a History of Math class. This makes the most sense to me since

  • solving polynomial equations (and the methods of reducing one type to another) plays an important role throughout the ages, but especially in 16th century mathematics.
  • students of the sciences certainly don’t need this info for practical reasons, so there’s little motivation to include it in Precalculus or the Calculus sequence.
  • that there is a general solution is enough to know for most upper level math classes, the actual methods and solutions being less important.
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    $\begingroup$ In particular, history of math class should deal with how Tartaglia/Cardano introduced the notion of imaginary numbers in order to get real solutions of the cubic. $\endgroup$ – Adam Dec 16 '18 at 22:19
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    $\begingroup$ The quartic solution formula looks like it belongs in abstract art. $\endgroup$ – Joshua Dec 17 '18 at 21:04
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    $\begingroup$ And should not be taught without scrupulously mentioning Abel's Theorem.\ $\endgroup$ – kcrisman Dec 18 '18 at 3:17
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I think it is very reasonable in grad abstract algebra to show that these formulas are corollaries of very natural manipulations of Lagrange resolvents. These natural manipulations also illustrate some other points, about "averaging" and Vandermonde determinants, and so on. All these pre-date Galois by decades, and were the context in which Galois, Abel, and Ruffini worked.

It is slightly strange that Lagrange resolvents are not visible in most "modern" treatments of Galois theory. At least one edition of van der Waarden's algebra books does treat them (following E. Noether?)

My own algebra notes (on-line at http://www-users.math.umn.edu/~garrett/m/algebra/ in chapter 23, section 3, as well as existing in a physical book) do in fact give more emphasis to Lagrange resolvents (which I think illustrate many interesting things) than to the fact that extensions are expressible in radicals if and only if the Galois group is solvable. This is certainly not the main point of field theory and Galois theory, in any case, but, rather, an interesting aside, in my opinion.

Use of Lagrange resolvents (and a little algebraic number theory) to express roots of unity in radicals is at http://www.math.umn.edu/~garrett/m/v/kummer_eis.pdf

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    $\begingroup$ Maybe worth linking the online notes? $\endgroup$ – Benjamin Dickman Dec 17 '18 at 2:52
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    $\begingroup$ @BenjaminDickman, done. $\endgroup$ – paul garrett Dec 17 '18 at 19:35
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I'm used to seeing them "starred" in an algebra 2 or "college algebra" course. Starred means the extra topics that very advanced classes could cover. My experience is even for those classes nobody bothers. I have taken a huge amount of engineering and science courses and never really felt the loss of these techniques.

The only time I solved cubics was in some of the harder AP chemistry stoichiometry problems where a cubic resulted in equilibrium problems. and then I just got x by approximation (guessing and revising guesses after feeding the number in, mechanically without a program or algorithm). And this worked fine for those chem problems (for example to get pH to single character after the decimal).

What you do see a hyooooge amount of the time is the quadratic. bsq-4ac needs to become iconic. It is in a gazzillion chem rate and equilibrium problems and it is all over physics and engineering. Either directly or in solving the characteristic equation of the 2nd order constant coefficient homo diffy Q (EE, controls/feedback, harmonic oscillator, quantum mechanics, etc.)

So I can tell you that I have "needed" the quadratic a metric gazillion amount of times. And the cubic almost never (and worked around by approximating). And really never solved a quartic.

Now I know you were asking about kids in math, but I just wanted to discuss why we don't bother with it in algebra 2 (where it sort of belongs by history and by topic). Because of course, the needs of math undergrads are irrelevent to a general algebra 2 class because these kids are a tiny fraction of kids in a high school class.

Now. The question becomes if you need to cover that stuff in other math classes. Personally I doubt it. Do you really encounter the need to analytically solve cubics or quartics in upper div math classes? No. And do you need it if you get a job that typical math grads do? (Actuary?) Probably not.

And in the tiny tiny fraction of the time that you do need it, you can just look it up in the "starred section" of your old book. Provided that you have strong manipulational muscles, you can just apply the formula for your specific problem then, without any qualms. But you really don't need to learn/remember it like you do the quadratic (not just knowing it, but using it readily).

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    $\begingroup$ "Because of course, the needs of math undergrads are irrelevent to a general algebra 2 class because these kids are a tiny fraction of kids in a high school class." This mass-production mentality is exactly what is wrong with so much of the math taught in schools today. We cater the education to the lowest quartile at the expense of crushing the spirit of the highest quartile. In the end, we're all dumber for it. $\endgroup$ – James S. Cook Dec 17 '18 at 17:06
  • $\begingroup$ All serious human endeavors in government and business require considering constraints and optimization. Less Rudin and a more operations research (a branch of math). Throw in some probability and statistics. And not measure theory but Box, Hunter, Hunter. (Like actually using the stuff!) $\endgroup$ – guest Dec 17 '18 at 17:54
  • $\begingroup$ Schools track all the time. It's a dirty word but it is clear that it happens and that it is a practical approach to the problem of different needs. It is very common in high school and is implicit in college (even including the don't go to college choice). It's a problem solving error here to assume that a change to the curriculum will apply to all students or even should. It shows a lack of awareness of even what goes on in schools. $\endgroup$ – guest Dec 17 '18 at 17:59
  • $\begingroup$ "Because of course, the needs of math undergrads are irrelevent to a general algebra 2 class because these kids are a tiny fraction of kids in a high school class." Education serves a societal need as well as individual needs. One social need is the identification and training of future scientists, engineers, medical practitioners, economists, teachers, etc. The education of the (not so) tiny fraction is a principal concern, not a marginal one. $\endgroup$ – Dan Fox Dec 18 '18 at 6:00
  • $\begingroup$ "The only time I solved cubics was in some of the harder AP chemistry stoichiometry problems where a cubic resulted in equilibrium problems." There's a cause and effect issue to note here - math and math-adjacent classes tend to ask problems that only need quadratics while avoiding cubics because they know those are what their students can solve. $\endgroup$ – Henry Towsner Dec 18 '18 at 14:14
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Abstract Algebra, under field extensions/Galois theory.

Unlike quadratics, there are relatively few applications (hard-applied or applications to other theory) where exact solutions to cubic or quartic matter, and no reason you wouldn't just ask a computer algebra system or consult a reference book if you need them, rather than trying to memorize or work things out by hand.

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It seems a mistake to include such formulas in middle school or high school education except perhaps in supplementary material or in classes directed at motivated and talented students. For the most part the interest of such formulas comes from number theoretic and algebraic considerations that are not of such broad interest or applicability. Their complexity is such that including them simply reinforces a problematic memorize formulas mentality already strong in students who can find the roots of a quadratic polynomial but have no idea how to complete the square.

It seems reasonable (although not necessary) to discuss such formulas in a calculus class as part of a broader discussion of root finding. It should be standard to include in calculus classes exercises directed at the localization of and determination of the number of roots of a cubic polynomial. Qualitative arguments such as this polynomial with leading coefficient $x^{3}$ has a unique local minimum at which it takes a negative value so has three real roots, one of which is greater than number at which the local minimum occurs are well within the reach of a basic calculus class. Bisection is generally taught in calculus classes as well (often under some name such as Bolzano's theorem, with a nonconstructive proof ...), and it would be a good thing to teach Newton's method too. It is instructive to combine such methods to find (by hand) approximate roots of a cubic polynomial and compare with what the solutions in radicals give. Such a comparison highlights that an expression in radicals may not be terribly useful in a scientific/numerical context, because there remains the need to approximate the radicals!

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It should be taught in high school after students learn solving quadratic equations. The difficulties with having to work with roots of negative numbers should not be an objection here, rather such problems should be used as an extra motivation to later learn about complex numbers. And later at university it can be revisited again to introduce Galois theory.

Invoking the complexity of the general solution is not a good argument either. Solving third and fourth degree equations when the equations are specified using numeric values can be done quite fast and efficiently using the methods that lead to the general solution, while using the general solution directly is quite awkward. This happens for the same reason why solving a system of linear equations using Gaussian elimination is quite efficient, while the general solution is massively complex even for systems as small as 4 equations with 4 unknowns. The complexity of the solution increases exponentially with the number of coefficients, but if the coefficients are just numbers, you can always add up many different terms to get to short expressions.

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I think it should be taught in first year Calculus.

If it is taught in high school, there is a good chance many of the students will never use it again (a high proportion do not go on to college), so that information will be wasted. Teaching it to students who are already in college or university increases the odds that this information will actually be used.

In first year Calculus, there is usually a brief review of some high school topics, trigonometry, algebra, etc, before charging ahead with the new material. Even if a derivation is not provided, or students are not required to work through a derivation, they should at least know these solutions exist.

From my own background (engineering), we started solving third degree polynomials in second term of Physical Chemistry, when we had to work with Van der Waals Equation, a third-degree polynomial. Today, most university students have access to powerful software products that solve such equations numerically in the blink of an eye, but I did not. Working through the solutions for a third or fourth degree polynomial may have been a little tedious with a hand calculator, but it was doable. Instead of hitting a brick wall, we at least had tools at hand to proceed to a solution.

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