What are some useful factoring tips and tricks for math contests and competitions?

Question

What are some useful advanced factoring tips and tricks for someone interested in participating in an olympiad style math contest?

I am already familiar with taking the common factor, grouping like terms, basic identities and factoring quadratics. I am looking for more obscure/niche identities and general tips on how to factorise difficult polynomials with one or more variables. I lack imagination and often find it difficult to factorise expressions by adding and subtracting the same term.

If there is any book you would suggest I purchase or any online resource I can check out please share them. If you have general tips concerning factoring methodology, that would also be very helpful.

Answer 1

A list of semi-useful identities and techniques:

• $a^2 - b^2 = (a-b)(a+b)$, this is applied the most often.
• $a^2 + 2ab + b^2 = (a+b)^2$, except if you want you can always introduce a term, such as $a^2 + b^2 = a^2 + b^2 + 2ab - 2ab = (a+b)^2 - 2ab$, if that would ever help you. The idea of introducing terms this way is super important and you should be on the look out for it, since it is one of the more creative ideas here.
• $(1 + x)^n = \sum_{k=0}^n x^k {n \choose k}$, where $n \choose k$ is the binomial coefficient. This is well known as the binomial theorem and often appears in problems of summations and coefficients. It should also be know that: $n(1+x)^{n-1} = \sum_{k=0}^n k x^k {n \choose k}$ by taking the derivative. Finally, if you plug $x = 1$ in, you can get rid of $x$ which can be occasionally useful for dealing with summations.
• $a^4 + 4b^4 = (a^2 + 2b^2 + 2ab)(a^2 + 2b^2 - 2ab)$, Sofia-Germaines identity. This is useful to memorize since the odds of you finding it on your own are next to none when time is on the clock.
• The famous HM-AM-GM-QM inequality, visible here.

I'm stopping the list here, as now I would recommend you go read a book, and practice problems. This is the best way to learn, and best way to practice. There are lots of great problem solving books, I think proofs from the book by Martin Aigner is a good, fun read with interesting ideas throughout the book, which contains many more theorems and identities. Specifically, for problem solving competitions, Putnam and Beyond is good if you are at that level, otherwise you should get one of the classic math olympiad books, which you can google. Finally, this site here may be of interest, it is a compilation of lecture notes for math olympiad. Good luck!