Useful Equalities

This are some very well known and useful equalities that are often applied when solving problems.

The difference between two powers:

$ a-b=a-b $
$ a^2-b^2=(a-b)(a+b) $
$ a^3-b^3=(a-b)(a^2+ab+b^2) $
$ a^n-b^n=(a-b)(a^{n-1}+a^{n-2}b+a^{n-3}b^2+...+a^2b^{n-3}+ab^{n-2}+b^{n-1}) $
$ a^n-b^n=(a-b)\sum_{k=0}^{n-1} a^{n-1-k}b^k $

The sum between two odd powers:

$ a+b=a+b $
$ a^3+b^3=(a+b)(a^2-ab+b^2) $
$ a^5+b^5=(a+b)(a^4-a^3b+a^2b^2-ab^3+b^4) $
$ a^{2n+1}+b^{2n+1}=(a+b)(a^{2n}-a^{2n-1}b+a^{2n-2}b^2-...+a^2b^{2n-2}-ab^{2n-1}+b^{2n}) $
$ a^{2n+1}+b^{2n+1}=(a+b)\sum_{k=0}^{2n}(-1)^ka^{2n-k}b^k $

The sum of two number at a power:

$ a+b=a+b $
$ (a+b)^2 = a^2 + 2ab+ b^2 $
$ (a+b)^3 = a^3 + 3a^2b+ 3ab^2 +b^3 $
$ (a+b)^n=a^n+ {n \choose 1}a^{n-1}b+{n \choose 2}a^{n-2}b^2+...+{n \choose {n-2}}a^2b^{n-2}+{n \choose {n-1}}ab^{n-1}+b^n $
$ (a+b)^n=\sum_{k=0}^n{n \choose k}a^{n-k}b^k $
$ (a+b)^n=a^n+ C_n^1a^{n-1}b+ C_n^2a^{n-2}b^2+...+C_n^{n-2}a^2b^{n-2}+C_n^{n-1}ab^{n-1}+b^n $
$ (a+b)^n=\sum_{k=0}^n  C_n^k a^{n-k}b^k $
$ C_n^k={n \choose k}=\frac{n(n-1)...(n-k+1)}{k!}=\frac{n!}{k!(n-k)!} $

This equlities are true only if $ ab=ba $ (in commutative rings).

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