Need resources for difficult Algebraic Identites.

Question

Where can I find all the algebraic identities and their proofs of following types,

• $a^3+b^3+c^3–3abc≡(a+b+c)(a^2+b^2+c^2–ab–bc–ca)$

• $a^3+b^3+c^3–3abc≡ \dfrac12 (a+b+c)((a-b)^2+(b-c)^2+(a-c)^2$

• $a+b+c=2s$ then $a^2+b^2–c^2+2ab=4s(s–c)$.

• If $a+b+c=0$, then $\left[ \dfrac{a}{b+c} + \dfrac{b}{a+c} + \dfrac{c}{a+b} \right] \left[\dfrac{b+c}{a} + \dfrac{a+c}{b} + \dfrac{a+b}{c} \right]=9$

There are many more similar difficult identities. I need a list of all of them with their proofs.

Any help will be appreciated, thanks in advance.

Answer 1

I've found one book out there which covers these type of identities and problem exercises -- SSC Elementary and Advanced Mathematics.