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I'm studying math on my own and never had university math training. I'm able to read undergraduate math (baby Rudin Analysis, Linear Algebra Done Right by Axler). Moreover, I can write proofs on most of what I read.

One problem, I'm not "comfortable" with basic or school algebra "tricks". For example:

$$ a^3 + b^3 = (a + b) (a^2 - ab +b^2) $$ $$ a'b' - ab = (a' - a)(b' - b) + a(b' -b) + b(a' - a)$$

While they are easy to see they are true, generally, they're not part of my tools when writing. From what I see, math uni students have it as a second nature, and they can use it instantly in their proofs.

I need to practice such problems within a short period. Is there a section of some book with exercises that can bring me to such level?

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Let me first make some comments on the two examples you gave.

The second example is not a known "trick" one would learn in high school. It is a calculus trick, if you can call it that, because it allows you to show that $a'b'$ is close to $ab$ if $a'$ is close to $a$ and $b'$ is close to $b$. So it's really something that would be devised specifically for that purpose, and one wouldn't typically see anything like that before, say, proving that $(x,y) \mapsto xy$ is a continuous function or proving the rule about the derivative of a product. I certainly haven't memorized this identity, but I know from experience (with calculus) that something along those lines is possible, and that's enough.

The first example is the special case of the identity $$x^n - y^n = (x - y)(x^{n-1} + x^{n-2}y + \dots + y^{n-1})$$ when $n = 3$, $x = a$ and $y = -b$. The general form of this identity is so well-known that if you wanted a list of high school algebra identities similar in importance to that one, the list might be five or ten items long at most. In that case, the other answer's suggestion of Algebra by Gelfand could be what you want.

However, if you want to develop a high level of skill in the manipulative aspects of algebra, there are a couple of sources I can recommend that are at an advanced high school level.

  • A Problem Book in Algebra, by Krechmar.

  • Higher Algebra: A Sequel to Higher Algebra for Schools, by Ferrar.

These books were formerly used by students preparing for the most competitive university entrance examinations in the USSR and the UK, respectively. The first book consists of problems with solutions, while the second is an actual textbook.

Edit: Here are two derivations of the second identity that might make it seem more natural. Assume $a'$ is close to $a$ and $b'$ is close to $b$. You are trying to prove $a'b'$ is close to $ab$, but there is no simple way to write the difference between them.

Method 1. Instead, write that $a'b'$ is close to $a'b$, which is in turn close to $ab$: $$a'b' - a'b = a'(b'-b)$$ $$a'b - ab = (a' - a)b$$

Adding these two equalities, we get $$a'b'-ab = a'(b' - b) + (a' - a)b.$$

The form that you have is obtained by replacing $a'(b' - b)$ with $a(b' - b)$, which is close to it, plus a correction term.

Method 2. Write $a' = a + \Delta a$ and $b' = b + \Delta b$. Then $$a'b' - ab = (a + \Delta a)(b + \Delta b) - ab = a \Delta b + b \Delta a + \Delta a \Delta b,$$ which is your identity.

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  • $\begingroup$ +1 --- I was about to comment that the first is a common and standard identity that, for those continuing in math, is akin to the multiplication table, whereas the second is more of a trick, and regarding things like the second, you pick those up when you see them used sufficiently often (and if you almost never encounter it, then you don't worry about it). Then I saw your answer, fortunately before I started writing . . . $\endgroup$ – Dave L. Renfro Jul 20 '18 at 12:36
  • $\begingroup$ "want to develop a high level of skill in the manipulative aspects of algebra". Yes, this is what I'm looking for. Still, @José Carlos Santos recommendation is a useful "list" of the well-known problems. (The two problems I picked are a random choice, but you provided a good explanation). $\endgroup$ – one1 Jul 20 '18 at 16:45
  • $\begingroup$ The Gelfand/Shen book is probably one of the best places for you to begin, given your background (because it's a relatively quick read and the focus is on mathematical reasoning and algebraic essentials), but it's not what I would recommend for someone who is specifically looking for a high level of skill in manipulation. For that, see the books I mention (and provide links to) in the paragraph beginning with In older advanced "school level" algebra texts from the 1800s in this answer of mine. $\endgroup$ – Dave L. Renfro Jul 21 '18 at 9:36
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Perhaps that Algebra, by Israel Gelfand and Alexander Shen, is what you're after.

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