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# 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}= a^{2}+ 2ab + b^{2} - (a − b)
^{2 }= a^{2}− 2ab + b^{2} - (a + b)
^{3}=a^{3 }+ b^{3 }+ 3ab(a + b) - (a – b)
^{3 }= a^{3 }– b^{3}– 3ab(a – b) - (a + b)
^{4}= a^{4}+ 4a^{3}b + 6a^{2}b^{2}+ 4ab^{3}+ b^{4} - (a − b)
^{4}= a^{4}− 4a^{3}b + 6a^{2}b^{2}− 4ab^{3}+ b^{4} - (a + b + c)
^{2}= a^{2}+ b^{2}+c^{2}+ 2ab + 2ac + 2bc - a
^{2}– b^{2}= (a + b)(a – b) - a
^{2}+ b^{2}= (a + b)^{2}- 2ab - a
^{3}– b^{3}= (a – b)(a^{2}+ ab + b^{2}) - a
^{3}+ b^{3}= (a + b)(a^{2}– ab + b^{2})

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

= 2^{3 }× 5

**Answer:** **The prime factorization of 40 is 2 ^{3 }× 5.**

**Example 2:** Factorize a^{2 }- 625.

**Solution: **

a^{2 }- 625 = a^{2 }- 25^{2 }

Using the known identity, we can factorize this polynomial

a^{2 }- 25^{2 } is of the form a^{2 }- b^{2}

We know that a^{2 }- b^{2 }= (a+b) (a-b)

Thus we factorize the polynomial as (a + 25) (a - 25)

**Answer: **a^{2 }- 625 is factorized as (a + 25) (a - 25)

**Example 3:** If the factorization of a number is 2^{2 }× 3^{2} × 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)

= 2^{2} × 3^{2} × 5

= 180

**Answer: ****The factorized number is 180.**

## FAQs on Factorization Formula

### What Does N Represent in Factorization Formula?

The general factorization formula is expressed as N = X^{a} × Y^{b} × Z^{c}. Here, N represents the factorized number.

### What Do X, Y, Z Represent in Factorization Formula?

The general factorization formula is expressed as N = X^{a} × Y^{b} × Z^{c}. Here, X, Y, Z represent the factors of a factorized number.

### What Do a, b, c Represent in Factorization Formula?

The general factorization formula is expressed as N = X^{a} × Y^{b} × Z^{c}. Here, a, b, c represent the exponential powers of the factors of a factorized number.

### How to Factorize the Given Algebraic Expressions?

We factorize the algebraic expressions using the known algebraic identities. x^{2 }+ 6x + 9 is factorized as (x+3)(x + 3) using the known algebraic identity (x+a) ^{2 }= x^{2} +2ax +a^{2}.

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