Factorization is a method of breaking down an entity into a product of another entity. It is simply the process of resolution of an integer or polynomial into factors such that when multiplied together results in the original polynomial or integer.
Method of Factorization is quite useful to reduce any algebraic expression or quadratic equation into simpler form rather than writing product of terms within brackets. In this article, you will get acquainted with the definition of Factorization, Necessary Factorization Formulas, Methods on How to Factorize with Examples, etc. all explained in detail.
What is meant by Factorization in Mathematics?
Factorization is nothing but writing the given expression in terms of the product of its factors and the factors can be either numbers, variables, or even algebraic expressions. To Factor is to break down a number into products of other numbers that when multiplied can result in the original number.
For Example Factorization of 36 is 9*4 where 9, 4 are the factors. Now, that you are aware of finding the factorization of numbers let us move ahead and determine the factorization of a quadratic polynomial.
Factorization in Algebra
Numbers 1, 2, 3, 4, 6, 8, 12, 24 are all factors of 24 as they divide 24 without leaving a remainder. This is a vital topic to be learned for learning other concepts like simplifying expressions, simplifying fractions, and solving equations. It is referred to as Algebra Factorization.
General Factorization Formulas List
- a2 – b2 = (a – b)(a + b)
- (a + b)2 = a2 + 2ab + b2
- (a – b)2 = a2 – 2ab + b2
- a3 – b3 = (a – b)(a2 + ab + b2)
- a3 + b3 = (a + b)(a2 – ab + b2)
- (a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca
- (a – b – c)2 = a2 + b2 + c2 – 2ab + 2bc – 2ca
Factorization Methods
Algebraic Expressions can be factorized using 4 different methods and we have stated all of them below for your knowledge. They are as follows
- Common factors method
- Regrouping terms method
- Factorization using identities
- Factors of the form (x+a) (x+b)
Factorization using Common Factors Method
In this method, we will simply find the common factors for each term of the given expression. For Example Factorization of 4x+16 can be written as 4(x+4). Here we have taken 4 i.e. common factor for both the terms
Factorization using Regrouping of Terms Method
Regrouping of Terms is a technique in which we will rearrange the given expression on basis of like and unlike terms.
For Example 6xy + 3x + 8y + 4. Rearranging them and expanding to get the factored form 6*x*y +3*x+8*y+4*1
= 3x(2y+1)+4(2y+1)
Now taking out the common factor (2y+1) we have the following terms as factors
= (3x+4)(2y+1)
Factorization using Identities
For Factorization using Identities we can factorize the algebraic expression as follows
Example: Factorize 4x2-16
Using Algebraic Identities we know a2 – b2 = (a – b)(a + b)
we can write 4x2-16 = (2x)2 – (4)2
= (2x+4)(2x-4)
Factorization of Factors in the form of (x+a) (x+b)
If a given expression is in the form of x2 + (a + b) x + ab then its factors can be written as (x+a) (x+b).
For Example: x2 + 7x + 12
We can rewrite the given expression as x2 + (4+3)x + 4.3
Now, after comparing with the general expression form we get
a+b = 7 ….(i)
ab = 12 ….(ii)
a = 4, b =3 both satisfies the given condition thus we can write the factors in the form of (x+4)(x+3)
Factorization Method Examples
Example 1.
Factorize the Quadratic Polynomial x2 + 5x + 6?
Solution:
Given Quadratic Polynomial = x2 + 5x + 6
We can rewrite the given quadratic polynomial as x2 + (2+3)x + 2.3
After comparing with the general expression we have
a+b = 5 ….(i)
ab = 6 ….(ii)
2, 3 satisfies the given quadratic polynomial when substituted so we can write the quadratic polynomial in the form of (x+2)(x+3)
Therefore, Quadratic Polynomial x2 + 5x + 6 when factorized results in (x+2)(x+3)
Example 2.
Factorize x2 – 16?
Solution:
Given Expression is x2 – 16
Factorizing the expression using identities we have a2 – b2 = (a – b)(a + b)
The above expression can also be written as (x)2 – (4)2 = (x-4)(x+4)
Example 3.
Factorize (8 x + 8 x3) + (x4 + x6)?
Solution:
Given Expression is (8 x + 8 x3) + (x4 + x6)
Now taking the common factors we get the equation as follows 8x(1+x2)+x4(1+x2)
= (8x+x4)(1+x2)
These are the required factors.
FAQs on Factorization
1. What is meant by Factorization?
Factorization is a technique of breaking down an entity into products of another entity.
2. What are the different methods of Factorization?
The different methods of Factorization are
- Common factors method
- Regrouping terms method
- Factorization using identities
- Factors of the form (x+a) (x+b)
3. What are Factorisation Formulas?
Formulas of Factorization are listed below
- a2 – b2 = (a – b)(a + b)
- (a + b)2 = a2 + 2ab + b2
- (a – b)2 = a2 – 2ab + b2
- a3 – b3 = (a – b)(a2 + ab + b2)
- a3 + b3 = (a + b)(a2 – ab + b2)
- (a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca
- (a – b – c)2 = a2 + b2 + c2 – 2ab + 2bc – 2ca