Methods of Factorization
Factorization is the process of breaking down an algebraic expression into the product of its factors. Different methods are used for factorization based on the type of algebraic expression. Below are the common methods of factorization:
1. Common Factor Method
In this method, we take out the greatest common factor (GCF) from the given terms. This is the simplest form of factorization.
2. Regrouping Method
This method is used when the expression has four or more terms. We group terms in pairs and factorize them separately.
3. Factorization by Using Identities
Factorization using identities involves recognizing standard algebraic formulas and applying them.
Important Identities:
- (a + b)² = a² + 2ab + b²
- (a – b)² = a² – 2ab + b²
- a² – b² = (a – b)(a + b)
4. Factorization of Quadratic Expressions
A quadratic expression is of the form ax² + bx + c. Factorization can be done using the splitting the middle term method.
Example:
Factorize x² + 5x + 6
- Find two numbers whose sum is 5 and product is 6 → (2 and 3)
- Split the middle term:
x² + 2x + 3x + 6 - Group and factorize:
x(x + 2) + 3(x + 2) - Take out the common binomial:
(x + 2)(x + 3)
5. Factorization by Splitting the Middle Term
This method is used for quadratic expressions where the middle term is split into two terms whose sum equals the middle coefficient and product equals the product of the first and last term.
Example:
Factorize 2x² + 7x + 3
- Product of 2 × 3 = 6, sum 7
- Numbers: 6 and 1
- Split the middle term:
2x² + 6x + x + 3 - Group and factorize:
2x(x + 3) + 1(x + 3) - Take out the common binomial:
(2x + 1)(x + 3)
6. Factorization by Taking Common Binomial Factors
If a binomial factor is common in all terms, it can be factored out.
Example:
Factorize (x + y)² – 5(x + y)
- Taking (x + y) common:
(x + y) [(x + y) – 5] - Final factorized form:
(x + y)(x + y – 5)
Conclusion
Factorization is a powerful algebraic tool that helps simplify expressions and solve equations. The choice of method depends on the type of expression, and mastering these techniques is essential for solving algebraic problems efficiently.