Methods of Factorization of Polynomial
There are four different factoring polynomials formulas as follows:
- Greatest Common Factor (GCF)
- Substitution Method
- Grouping Method
- Difference of Two Squares Identity
Greatest Common Factor (GCF)
This is the basic method for factoring polynomials as Greatest Common Factor (GCF). In this method, we have to identify any common factor in all the terms. If any common factor is found then factor it out of the polynomial. It is simply a reverse procedure of the distributive law.
In the case of distributive law, we get:
a(b+c) = ab + ac
Whereas in the case of factorization, we invert the process
ab + ac = a(b+c)
Here ‘a’ is the greatest common factor.
Example: Factorize 2x2 + 4x
Solution:
= 2x2 + 4x
= 2(x2 + 2x)
Substitution Method
This method can be very helpful if a given polynomial is too complex, in such case, we have to figure it out and try substituting the complicated terms with a simpler term to solve. Therefore it makes it much easier to factor out.
Example: Factorize 3x2 + 12xy + 9y2
Solution:
Let’s substitute A = 3x and B = 3y
3x2 + 12xy + 9y2
= (A)2 + 2AB + (B)2
= A2 + 2AB + B2
= (A + B)2
= (3x + 3y)2
Grouping Method
If an expression has an even number of terms but no common factors exist for any of the terms, we can pair the terms together and get the common factor for each pair:
Example: Factorize 3x + 4ay – 3y – 4ax
Solution:
= 3x – 3y + 4ay – 4ax
= 3(x – y) + 4a(y – x)
= 3(x – y) – 4a(x – y)
= (3 – 4a)( x – y)
Difference of Two Squares Identity
This is a specific technique used when dealing with polynomials that can be expressed as the difference of two perfect squares. The identity states that the expression a2 – b2 can be factored as (a + b)(a – b).
Example: Factorize: 16x2 – 25
Solution:
= 16x2 – 25
= (4x)2 – (5)2
Comparing with a2 – b2 = (a – b)(a + b)
= (4x + 5)(4x – 5)
Polynomials – Definition, Standard Form, Types, Identities, Zeroes
Polynomials: In mathematics, polynomials are mathematical expressions consisting of indeterminates (also called variables) and coefficients, that involve only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. They are used in various fields of mathematics, astronomy, economics, etc. There are various examples of the polynomials such as 2x + 3, x2 + 4x + 5, etc.
In this article, we will learn about, Polynomials, Degrees of Polynomials, Examples of Polynomials, Zeros of Polynomials, Polynomial Equations, and others in detail.
Table of Content
- What are Polynomials?
- Polynomials Definition
- Polynomials Examples
- Characteristics of Polynomials
- Standard Form of a Polynomial
- Degree of a Polynomial
- Degree of Single Variable Polynomial
- Degree of a Multivariable Polynomial
- Terms in a Polynomial
- Types of Polynomials
- Properties of Polynomials (Theorems of Polynomials)
- Operations on Polynomials
- Addition of Polynomials
- Subtraction of Polynomials
- Multiplication of Polynomials
- Division of Polynomials
- Factorization of Polynomials
- Methods of Factorization of Polynomial
- Greatest Common Factor (GCF)
- Substitution Method
- Grouping Method
- Difference of Two Squares Identity
- Zeros of Polynomial
- How to Find Zeros of Polynomials?
- Linear Polynomial
- Quadratic Polynomial
- Cubic Polynomial
- Higher Degree Polynomial
- Polynomial Identities
- Polynomial Equations
- Solving Polynomials
- Polynomial Functions
- Polynomials Class 9 Extra Questions
- Polynomials Class 10 Extra Questions
- Practice Problems on Polynomial
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