Which difficulty level would this problem be considered?

[CynicusRex]

https://www.physicsforums.com/threads/discriminant-of-cubic-equation-in-terms-of-coefficients.715480

The thread is really old and didn't want to post this trivial question in there.

The problem is: A cubic equation x³ + px² + qx + r = 0 has three different roots x₁, x₂, x₃.
Find (x₁-x₂)²(x₂-x₃)²(x₁-x₃)² as an expression containing p, q, r. This polynomial p, q, r is called the discriminant of the cubic equation.

I'm asking because obviously I wasn't able to solve this problem and it's in a book mentioned in the thread: how to self study high school mathematics with the following as part of the description: "This book should be ideal for people new to algebra, or people who find that they remember very little of their algebra classes.

The introduction of the book also mentions: "However, if you have difficulties solving a problem (and some of them are quite difficult), you may read the hint or start to read the solution. If there is no solution in the book for some problem, you may skip it (it is not heavily used in the sequel) and return to it later.

I guess this is one of the more difficult ones? Either way, when is this taught?

PS I wanted to specify the title, but can't change it.

 

[John Park]

It could be high school (in some part of the world). How far did you get? Did you understand the significance of the three roots x₁, x₂, x₃?

 

[mathwonk]

this sort of thing is a very, very tedious exercise in computing symmetric functions. i would never recommend this as a problem to anyone. that said, there is an algorithm for expressing any symmetric function in terms of the elementary ones, and in principle this will yield to that process. you might start with the simplifying assumption that the sum of the roots is zero, i.e. that p=0.

This problem is discussed in great detail on pages 12-24 of my (graduate) algebra notes, where i use the computer program mathematica for the crude heavy lifting, and then explain some clever ways to do it without that assistance. By the way since you asked about difficulty level, it depends on your meaning of "difficulty". I.e. since there is an algorithm to do it, if done that way, absolutely no cleverness or originality is required, yet there is a great deal of tedious calculation which is very easy to get wrong and hard to complete successfully, so the "difficulty" level in one sense is quite large, but not so in creativity level. Clever people however usually find a clever way to do anything, and a colleague of mine contributed such a solution to this problem at least in the reduced case.

Notice the reason I explained this topic in such detail in my notes was exactly that I encountered so many books where it was left as an "exercise" to the reader, I suspect because it is actually so tedious to do. I myself had a great deal of difficulty just computing it correctly by hand. In fact I think I never completed it. So in my notes I bashed it out in the long tedious way using mathematica, and then also explained more realistic ways to do it. When I wrote these notes I was committed to not deceiving the reader as to how hard something was by leaving it as an essentially undoable "exercise".

http://alpha.math.uga.edu/%7Eroy/844-2.pdf

 

[mpresic]

The answer is it generally is not taught. I am told that in the old days > 50 years ago, they taught a method for solving (the general) cubic equations, but I'm not sure I believe it. I think a very good junior/senior undergraduate (before college graduation) might get to it. However even very good undergraduate courses in abstract algebra often put their priority in different sets of algebraic structures and problems, depending on the professors taste.

For myself, I learned the method from the (Schaum) Mathematical Handbook perhaps 20 years after I completed my Masters in Physics. There are professional mathematicians who may not know the method, and do not feel poorer for it.

It is interesting that the general cubic equation was not solved until the (sixteenth) century by Italian mathematicians.

 

[mpresic]

I should add that the solution in the Math handbooks are a method, not a solution in terms of p, q, r. The only solution I know of in general p,q,r was generated by a (my) computer algebra routine in MATLAB. The solution for the three roots together goes multiple pages.
I would not be too surprised in the answer in terms of p, q and r was never done before computer algebra, since the solution is not very illuminating.