Factoring: Some special cases

Here are some algebraic formulas, which can be used for factoring and simplifications of expressions. All of them can be proved disclosing of brackets and collecting like terms of items.

The binomials formulas

(a + b)2 = a2 + 2ab + b2	– square of the sum
(a – b)2 = a2 – 2ab + b2	– square of the difference
a2 – b2 = (a – b)(a + b)	– difference of squares
(a + b + c)2 = a2 + b2 + c2 + 2ab + 2ac + 2bc

The trinomials formulas

(a + b)3 = a3 + 3a2b + 3ab2 + b3	– cubes of the sum
(a – b)3 = a3 – 3a2b + 3ab2 – b3	– cubes of the difference
a3 + b3 = (a + b)(a2 – ab + b2)	– sum of cubes
a3 – b3 = (a – b)(a2 + ab + b2)	– difference of cubes

The formulas for the fourth degree

(a + b)4 = a4 + 4a3b + 6a2b2 + 4ab3 + b4

(a – b)4 = a4 – 4a3b + 6a2b2 – 4ab3 + b4

a4 – b4 = (a – b)(a + b)(a2 + b2)

Binomial theorem

(a + b)n = an + nan – 1b + n(n – 1)2an – 2b2 + ... + n!k!(n – k)!an – kbk + ... + bn

(a - b)n = an - nan – 1b + n(n – 1)2an – 2b2 + ... + (-1)kn!k!(n – k)!an – kbk + ... + (-1)nbn