Factorization Using Identities

Factorization is the process of breaking down an algebraic expression into the product of simpler expressions. Algebraic identities play a crucial role in simplifying and factorizing expressions.

In this section, we will focus on three important identities:

1. (a + b)² = a² + 2ab + b²
2. (a – b)² = a² – 2ab + b²
3. a² – b² = (a – b)(a + b)

These identities help in quickly factorizing quadratic and binomial expressions without lengthy calculations.

1. Factorization Using (a + b)² = a² + 2ab + b²

Theory:

This identity represents the expansion of a binomial squared. It is useful when we have a quadratic expression in the form a² + 2ab + b², which can be rewritten as the square of a binomial.

Example 1: Factorize x² + 6x + 9

Step 1: Identify terms:

• The first term x² is (x)²
• The last term 9 is (3)²
• The middle term 6x is 2 × x × 3

Since it matches the identity (a + b)² = a² + 2ab + b², we can rewrite it as:
(x + 3)²

Thus, x² + 6x + 9 = (x + 3)²

Example 2: Factorize 4y² + 12y + 9

Step 1: Identify terms:

• The first term 4y² is (2y)²
• The last term 9 is (3)²
• The middle term 12y is 2 × (2y) × 3

Since it matches the identity (a + b)² = a² + 2ab + b², we can rewrite it as:
(2y + 3)²

Thus, 4y² + 12y + 9 = (2y + 3)²

2. Factorization Using (a – b)² = a² – 2ab + b²

Theory:

This identity represents the expansion of a squared binomial with subtraction. It is used when we have an expression in the form a² – 2ab + b², which can be rewritten as the square of a binomial with subtraction.

Example 1: Factorize y² – 10y + 25

Step 1: Identify terms:

• The first term y² is (y)²
• The last term 25 is (5)²
• The middle term -10y is 2 × y × (-5)

Since it matches the identity (a – b)² = a² – 2ab + b², we can rewrite it as:
(y – 5)²

Thus, y² – 10y + 25 = (y – 5)²

Example 2: Factorize 9m² – 12m + 4

Step 1: Identify terms:

• The first term 9m² is (3m)²
• The last term 4 is (2)²
• The middle term -12m is 2 × (3m) × (-2)

Since it matches the identity (a – b)² = a² – 2ab + b², we can rewrite it as:
(3m – 2)²

Thus, 9m² – 12m + 4 = (3m – 2)²

3. Factorization Using a² – b² = (a – b)(a + b)

Theory:

This identity is known as the difference of squares and is used when an expression is in the form a² – b². Instead of expanding, this identity helps us directly factorize the given expression into the product of two binomials.

Example 1: Factorize x² – 25

Step 1: Identify terms:

• The first term x² is (x)²
• The second term 25 is (5)²

Since it matches the identity a² – b² = (a – b)(a + b), we can rewrite it as:
(x – 5)(x + 5)

Thus, x² – 25 = (x – 5)(x + 5)

Example 2: Factorize 4x² – 9y²

Step 1: Identify terms:

• The first term 4x² is (2x)²
• The second term 9y² is (3y)²

Since it matches the identity a² – b² = (a – b)(a + b), we can rewrite it as:
(2x – 3y)(2x + 3y)

Thus, 4x² – 9y² = (2x – 3y)(2x + 3y)