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# Factorization

Factorization can be understood easily if you are familiar with word ‘factor’. When we multiply any two Numbers and get third as its answer then those two numbers are termed as factors of resultant number. In simple words, factorization is defined as decomposition of things or an object into several products of another thing. In Math, factorization means crumbling of a number into a product of another objects, or factors which we multiply together gives original number.

Factorization can be performed on simple numbers as well as of Polynomials. For instance, number 18’s prime factors are 2 * 3 * 3, and polynomial y2 − 4 factors can be written as (y − 2) (y + 2). In all cases, i.e. of simple numbers and polynomials we get result as a product of simpler objects.
While finding factors or doing factorization our main desideratum is of reducing something to basic building blocks like when we factor Integer our aim is prime factorization and when we factor polynomial our aim is to get irreducible polynomial. While we study fundamental theorem of arithmetic we learn factoring of numbers or factoring integers. Similarly when we learn fundamental of Algebra we understand factoring of polynomial. Reverse process of polynomial factorization is called as expansion. Some examples are given below for better explanation of this concept.

## Factorize  the equation y4 – 81?

We know  the identity (x– y2) = (x + y) * (x – y),

Now here we will  look at the polynomial y4 – 81 and write it as :

Y4 – 34,

= (y2 – 32) * (y2 + 32),

Again looking at (y2 – 32), it can be written as (y – 3) * (y + 3),

So we say that the expression y4 – 81 will be factorized into :

(y – 3) * (y + 3) * (y2 + 32).

## Factorize 3x2 + 27?

Here we have 3 as common factor, so we will take out 3 common from both terms of the polynomial and we get:
= 3 * (x2 – 9) = 3 * (x2 – 32),
Now we know the formula for a2 – b2 = (a + b) * (a – b), so we will apply the above identity  and we get :
= 3 * (x – 3) * (x + 3).

## Factorize the polynomial: 5x2 + 30 * x +  45?

The above polynomial will be  solved by the method of splitting.
First we will take 5 as the common factor of the three terms, so the polynomial will be  written as follows:
= 5 *  (x 2 + 6 * x + 9),
= 5 *  (x 2  + 6 * x + 32),
= 5 *  (x 2  +  2 . 3 * x + 32),
Now we observe that the above expression is similar to the identity: (x + y)2 = (x2 + y2 + 2 * x * y).
So it can be written as:
= 5 * (x + 3)2.

## Find the factors of the polynomial given below:  16 x2 + 9 y2 – 24 * x *  y?

The above given polynomial can be written as:
(4 * x)2 + (3y)2 – 2 * 3 * 4 * x * y,
Now we observe that the above given polynomial resembles the identity
(a – b)2 = a2 + b2 – 2 * a * b,
So we  write the factors of the polynomial = (4x – 3y)2

## Factorize a(a – 1) – b (b – 1)?

The above given polynomial can be written as a2 – a – b2 + b,
= (a2 – b2) – (a + b),
= (a + b) * (a – b) – (a + b),
Taking (a + b) common, we get:
(a + b) * (a – b – 1).