What algebraic manipulations are needed for real analysis?

Question

What level of algebraic manipulations are needed for real analysis and higher math, and how should one attain it?

I'm able to solve most proofs in typical Real Analysis textbooks or university exams. Yet, despite that, when presented with a challenging, but elementary, algebraic manipulation (i.e. the kind seen on olympiads), I sometimes struggle.

Take, for example, some of the problems on this AoPS Intermediate Algebra Test, such as:

• Find all solutions (real and complex) to $\sqrt{x − 5} + \sqrt{x + 15} = 10$
• and to $\sqrt[3]{x^2 - 1} + \frac {20}{\sqrt[3]{x^2 - 1}} = 12$
• Simplify $\sqrt[4]{161 − 72\sqrt 5}$

or this one from Gelfand's Algebra:

• How to factor $a^{3} + b^{3} + c^{3} - 3abc$ into a product of polynomials

I find these and similar problems quite challenging.

Should I go back and build up these elementary algebraic manipulation skills before proceeding further with higher math? If so, how? When and how did you built these skills?

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In the spirit of math.SE, I'll share my "work" on this question so far:

1. It would seem yes, go back and build these foundations before proceeding, because you need the foundations before building.
2. But: The fact that you're able to succeed at math well beyond that raises doubts if these really are foundations, or more contest style challenges.
3. These algebraic manipulations, beyond the very basics, don't seem to be covered in any standard text, at any level, except for contest math. High school algebra and precalculus texts do not teach the advanced manipulations needed to solve problems like the above. And university level analysis, linear algebra, abstract algebra start after them.
4. All of which suggests that this level of manipulation is primarily part of contest math, but not generally a foundation or component of "standard" math. It's well established the difference: contest math revolves around "tricks," usually to remove deliberate obfuscation; standard math revolves around underlying concepts and techniques which expose a unity and clarity
5. Support for the above: When looking for resources on problems like the above, the results are almost entirely contest math sites (even the names, like "Simon's Favorite Factoring Trick," are from contest math).
6. Still: Eventually, students of "standard" math need to be able to use the techniques, eventually. There have been problems, such as What is the locus of points in the plane $\{v : v \cdot (v-a) = 0\}$ for fixed $a$? or simplfying $xy−bx−ay−ab=0$, where I get stuck on the manipulations. Do they just pick them up somehow? Do they become obvious once you've learn enough e.g. analysis and algebra? Update: Another example of where this came up in analysis is How to solve a particular system of non-linear multivariate equations?

To fill this gap, I invested some effort in going through books and resources on algebraic manipulations for contest math, and while I found it helpful, it certainly doesn't provide the satisfaction or insight as e.g. proving problems in analysis. So I'm confused whether I should return to analysis or continue working these manipulations, and, if I do, when and how can I pick these manipulations up? When and how have others done so?

Answer 1

Hopefully you have an advisor you can talk to about all these things, because in the limited channel of an MSE textbox there's only so much one can say. But, I'll say a few things.

Don't overthink this. In particular, don't obsess about contest math. Instead, follow what interests you... if you are interested in analysis, study analysis.

Those specific bulleted algebra problems you list are not particularly relevant to analysis. Algebra issues do come up, like the links in item 6, but deal with those issues as they arise.

Motivated by those links in item 6, I will add this. If there is one sub-topic within algebra that will be useful in more advanced analysis, it's Linear Algebra: eventually you will need to learn about vector spaces and inner products and norms and linear functionals and dual spaces and so on. But still, all of that only comes later in analysis. So I'll circle back to saying "follow what interests you..."