What is Factorization and its identities?

Factorization is the mathematical process of expressing an algebraic expression as a product of two or more factors. Below are the 15 essential identities and methods used to break down expressions:

Basic Methods & Linear Identities

$ab + ac = a(b + c)$
$ab - ac = a(b - c)$
$ac + ad + bc + bd = (a + b)(c + d)$

Square-Based Identities

$a^2 - b^2 = (a + b)(a - b)$

$a^2 + 2ab + b^2 = (a + b)^2$

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

$x^2 + (a + b)x + ab = (x + a)(x + b)$

$a^2 + b^2 + c^2 + 2ab + 2bc + 2ca = (a + b + c)^2$

$a^2 + b^2 + c^2 + 2ab - 2bc - 2ca = (a + b - c)^2$

Cubic & Advanced Identities

10. $a^3 + 3a^2b + 3ab^2 + b^3 = (a + b)^3$

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

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

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

14. $a^3 + b^3 + c^3 - 3abc = (a + b + c)(a^2 + b^2 + c^2 - ab - bc - ca)$

15. If $a + b + c = 0$ , then $a^3 + b^3 + c^3 = 3abc$

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