Class 8 - Factorisation Using Identities

Factorisation is the process of writing an algebraic expression as a product of its factors. One of the most useful methods of factorisation uses algebraic identities. This method is especially useful for factorising quadratic expressions and simplifying complex algebraic expressions. It is much faster than trial and error once you learn to recognise the patterns.

Table of Contents

• What is Factorisation Using Identities?
• Solved Examples On Factorisation Using Identities

 

What is Factorisation Using Identities?

Factorisation using identities means recognising that an expression matches the expanded form of a known identity and then rewriting it as the product form of that identity.

• Identity I: a² + 2ab + b² = (a + b)²

• Identity II: a² − 2ab + b² = (a − b)²

• Identity III: a² − b² = (a + b)(a − b)

• Identity IV: x² + (a + b)x + ab = (x + a)(x + b)
  
  

Read more:

• Important Questions on Factorization - Class 8
• Quadratic Equations
  
  

Solved Examples On Factorisation Using Identities

Example 1: Using (a + b)²

Problem: Factorise x² + 6x + 9.

Solution: Identify the pattern:

x² = (x)² 

9 = (3)² 

6x = 2 × x × 3 = 2ab 

a² + 2ab + b² = (a + b)²

• a = x, b = 3

Answer: x² + 6x + 9 = (x + 3)².

Example 2: Using (a − b)²

Problem: Factorise 4x² − 20x + 25.

Solution: Identify the pattern:

4x² = (2x)²

25 = (5)²

20x = 2 × 2x × 5 = 2ab 

Middle term is negative use (a − b)²

a² − 2ab + b² = (a − b)²

a = 2x, b = 5

Answer: 4x² − 20x + 25 = (2x − 5)².

Example 3: Using a² − b²

Problem: Factorise 9x² − 16.

Solution: Identify the pattern:

9x² = (3x)²

16 = (4)² 

Difference of two squares use a² − b² = (a + b)(a − b)

With a = 3x, b = 4:

9x² − 16 = (3x + 4)(3x − 4)

Answer: 9x² − 16 = (3x + 4)(3x − 4).

Example 4: Using (x + a)(x + b)

Problem: Factorise x² + 7x + 12.

Solution:

Find two numbers that:

• Add up to 7 (coefficient of x)

• Multiply to give 12 (constant term)

The numbers are 3 and 4:

3 + 4 = 7

3 × 4 = 12

Answer: x² + 7x + 12 = (x + 3)(x + 4).

Example 5: Common factor first

Problem: Factorise 3x² − 27.

Solution:

Step 1: Take out common factor: 3x² − 27 = 3(x² − 9)

Step 2: Apply a² − b² to (x² − 9): x² − 9 = x² − 3² = (x + 3)(x − 3)

Answer: 3x² − 27 = 3(x + 3)(x − 3).