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Whenever I teach multivariable calculus I find students really struggle with both finding critical points and the method of Lagrange multipliers. I think that the reason is the same: solving systems of polynomial equations requires different tactics and much more care than solving one variable polynomials, or linear systems. As far as I know, the students have no practice until they see it in the calculus context and even worse none of the books that I've seen even mention it.

Are there any books/resources/modules that deal with solving systems of polynomial equations?

I would also be interested in anyone who teaches algebra/precalculus weighing on if/how solving systems of polynomial equations is presented.

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    $\begingroup$ Just a suggestion: when you present, say, a Lagrange multiplier example in class for the first time, work out all the details yourself ahead of time. Then, present just the algebraic problem to be solved, devoid of the new concepts (Lagrange). Show the students how to solve that system. Then, present the new concepts and when you've set up the system, say, "Now is when we would do what we did before." Essentially, try to avoid the cognitive load of doing something new algebraically inside of doing something else entirely new to them. $\endgroup$
    – Brendan W. Sullivan
    Commented Mar 1, 2018 at 23:23
  • $\begingroup$ This is a really good suggestion: In the past I've tried to carefully explain the steps while doing the first couple of problems but I agree that the cognitive load is too great: in addition to learning new algebra techniques and new calculus techniques they have to differentiate between the two. $\endgroup$
    – Nate Bade
    Commented Mar 1, 2018 at 23:36

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I have found the following online resource from West Texas A&M University: College Algebra Tutorial 52: Solving Systems of Nonlinear Equations in Two Variables. It gives a step-by-step approach with examples, using either substitution or elimination. This could be useful for your students.

(For a more advanced perspective, see books such as Ideals, Varieties, and Algorithms by Cox, Little & O'Shea and Solving Systems of Polynomial Equations by Sturmfels.)

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    $\begingroup$ This is exactly the kind of thing I'm looking for, thank you so much! $\endgroup$
    – Nate Bade
    Commented Mar 4, 2018 at 18:51
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My college algebra text has section of solving systems of quadratics (intersections of ellipses, intersection of line and parabola, etc.) There is both calculational method and graphing method (numerical approximation). It is "starred" though and I don't really remember using it much in my life, even though I freaking covered it!

Not an iconic method like the quadratic equation itself or like systems of linear equations. My advice is to just plow through it. There is so much else going on within the grabbag of Calc 3. Div, grad, curl and all that.

The kids who are manipulational whizzes will do well (even if they don't remember their starred section of the College Algebra text, they can just handle it in context of new work) and those who aren't won't but what can you do. Lots of that going around anyways.

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    $\begingroup$ What text did you use? I'd be interested in taking a look. $\endgroup$
    – Nate Bade
    Commented Mar 1, 2018 at 22:53
  • $\begingroup$ @nate...I wonder if he used the textbook “Div Grad Curl and All That” by Schey. $\endgroup$
    – user52817
    Commented Mar 2, 2018 at 0:02
  • $\begingroup$ Hart College Algebra: amazon.com/College-algebra-William-United-Institute/dp/… Has a nice little chapter on investment calculations (great applied use of exponents) also. $\endgroup$
    – guest
    Commented Mar 2, 2018 at 0:04
  • $\begingroup$ When I teach this class, a theme is that multivariable calculus reduces many problems to “just algebra.” Sometimes it is linear algebra, and then you can had it off to a friend taking matrix theory. Sometimes it reduces to non-linear systems as in your example. $\endgroup$
    – user52817
    Commented Mar 2, 2018 at 0:07
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    $\begingroup$ Reminds me of a story from TAing chem. Had a smart but frustrated freshman in working problems. At one point, she told me "I just realized this whole subject is just an algebra class". I actually agree that it is mostly applied math and don't mind it (unlike chem profs who want to emphasize the concepts) but I still responded with some weasel statement. She looked me deadeye and said "Nope. It IS just algebra." And then she went from a low B to a strong A. Like a light switch! $\endgroup$
    – guest
    Commented Mar 2, 2018 at 0:44
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Solving systems of polynomial equations is hard in general. The examples in the textbook are specially cooked up to be possible. I am not sure that developing skill at solving such systems is a good use of ones time, especially in a course with as much conceptual content to master as Multivariable Calculus. Maybe let them get approximate solutions by graphing?

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  • $\begingroup$ I agree that it's hard, and probably not something we should take a week on. That said, right now we expect them do it without giving them any framework or resources on how. I feel like every year I watch students just get lost in the algebra here and I'm just surprised that it's not address in any of the texts I've seen. $\endgroup$
    – Nate Bade
    Commented Mar 1, 2018 at 20:19
  • $\begingroup$ Is this course coordinated at your institution? Can you simply change your expectation for these problems, so that they do not have to solve these systems anymore? $\endgroup$
    – Steven Gubkin
    Commented Mar 1, 2018 at 20:34
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    $\begingroup$ Changing the expectations has been a discussion in courses I've taught, but it usually come back to "well, at the end of the day, we want the students to be able to solve Lagrange multiplier problems." I suppose we could just give them simple ones but it seems like even the easiest require some touchy algebra. Take for example "Find the dimensions of the box with largest volume if the total surface area is 64 cm$^2$" from Pauls Notes. It doesn't seem like there's a middle ground between trivial and complicated. $\endgroup$
    – Nate Bade
    Commented Mar 1, 2018 at 22:01
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    $\begingroup$ @NateBade When it comes down to it, you are only really going to be able to solve these problems if something magically factors "easily" or if you end up with just quadratics. So "we want them to be able to solve Lagrange Multiplier problems" really becomes "we want them to be able to solve these incredibly specific problems which happen to be solvable". I cannot kick the feeling that we are mostly selling a lie in these courses. Compute thousands of integrals without ever meeting one without an elementary antiderivative, and you start to think you can integrate everything symbolically,etc $\endgroup$
    – Steven Gubkin
    Commented Mar 2, 2018 at 0:01
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    $\begingroup$ You could also teach them how to use Newton's method to solve the systems of equations approximately. $\endgroup$
    – Brian Borchers
    Commented Mar 2, 2018 at 15:15

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