Factorization Formula

Factorization is the process of rewriting an algebraic expression as a product of simpler factors. It makes expressions easier to solve, simplify, and work with.

Common Factorization Formulas

Some of the most commonly used factorization identities are:

a² − b² = (a − b)(a + b)

a² + 2ab + b² = (a + b)²

a² − 2ab + b² = (a − b)²

a³ + b³ = (a + b)(a² − ab + b²)

a³ − b³ = (a − b)(a² + ab + b²)

Example of Factorization

Consider the expression:

x² − 25

This is a difference of two squares because 25 = 5². Using the formula a² − b² = (a − b)(a + b):

x² − 25 = (x − 5)(x + 5)

So, the factors of x² − 25 are (x − 5) and (x + 5).

Factorization of a Quadratic Expression

For a quadratic expression such as x² + 5x + 6, find two numbers whose product is 6 and whose sum is 5. These numbers are 2 and 3.

Therefore:

x² + 5x + 6 = (x + 2)(x + 3)

Why Factorization Is Useful

Factorization can help simplify algebraic expressions, solve quadratic equations, reduce fractions, and identify the roots of polynomial equations.