Factorization Formula

The factorization formula is used to factorize a number. Factorization is defined as breaking an entity into a product of another entity, or factors, which when multiplied together give the original number. The factorization method uses basic factorization formula to reduce any algebraic or quadratic equation into its simpler form, where the equations are represented as the product of factors instead of expanding the brackets. The factors of any equation can be an integer, a variable, or an algebraic expression itself. Let us learn more about the factorization formula using solved examples in the following sections.

What is Factorization Formula?

The factorization formula factorizes a number quickly into smaller numbers or factors of the number. A factor is a number that divides the given number without any remainder. The factorization formula of a given value can be expressed as,

where,

• N = Any number
• X, Y, and Z = Factors of number N
• a, b, and c = exponents of factors X, Y, and Z respectively.

Factorization Definition

In math, factorization can be defined as the process of breaking down a number into smaller numbers which when multiplied together arrive at the original number. These numbers are broken down into factors or divisors. For example, 12 can be broken down as 3 × 4 and these two numbers are called factors.

List of Factorization Formulas for Algebraic Equation

There are many algebraic identities that help us in the factorization of algebraic expressions and the factorization of quadratic equations. Here are listed a few.

• (a + b)² = a² + 2ab + b²
• (a − b)² = a² − 2ab + b²
• (a + b)³ =a³ + b³ + 3ab(a + b)
• (a – b)³ = a³ – b³ – 3ab(a – b)
• (a + b)⁴ = a⁴ + 4a³b + 6a²b² + 4ab³ + b⁴
• (a − b)⁴ = a⁴ − 4a³b + 6a²b² − 4ab³ + b⁴
• (a + b + c)² = a² + b² +c² + 2ab + 2ac + 2bc
• a² – b² = (a + b)(a – b)
• a² + b² = (a + b)² - 2ab
• a³ – b³ = (a – b)(a² + ab + b²)
• a³ + b³ = (a + b)(a² – ab + b²)

Let us take a look at a few examples to understand the factorization formula.

Examples Using Factorization Formula

Example 1: Sam wants to factorize number 40. What the prime factorization of 40? Solve it by using the factorization formula.

Solution:

To find: Prime factorization of 40.
Using Factorization Formula,
Factorization Formula for any number, N = X^a × Y^b × Z^c
40 = 2 × 2 × 2 × 5
= 2³ × 5

Answer: The prime factorization of 40 is 2³ × 5.

Example 2: Factorize a² - 625.

Solution:

a² - 625 = a² - 25²

Using the known identity, we can factorize this polynomial
a² - 25² is of the form a² - b²

We know that a² - b² = (a+b) (a-b)
Thus we factorize the polynomial as (a + 25) (a - 25)

Answer: a² - 625 is factorized as (a + 25) (a - 25)

Example 3: If the factorization of a number is 2² × 3² × 5. Find the number by using the factorization formula.

Solution:

To find: The factorized number
Given:
The factorization of a number = 2 × 2 × 3 × 3 × 5.
Using the Factorization Formula,
Factorization Formula for any number, N = X^a × Y^b × Z^c
= (2 × 2 × 3 × 3 × 5)
= 2² × 3² × 5
= 180

Answer: The factorized number is 180.