Class 8 - Factorisation Using Common Factor

Factorisation is the most widely used concept in the different fields of Mathematics. To find the factors of a polynomial expression, the dimensions of a rectangular field expressed in terms of quadratic expressions, solving the values of the unknown from binomial and trinomial equations are a few examples where we use the concept of factorisation.

• Factors: A factor is a number or a variable that is multiplied by another number or variable to get a product or an expression.
• Common factors: The factors that are common to two or more given numbers are called common factors of the given numbers.
• An algebraic expression can be expressed as a product of two or more expressions, each of which is called a factor of the given expression.

The process of finding two or more expressions whose product is the given expression is called factorisation.

Table of Contents

• What is Factorisation by Common Factor?
• Solved Examples on Factorisation by Common Factor

What is Factorisation by Common Factor?

 Factorisation by common factor means rewriting an algebraic expression as a product of its highest common factor (HCF) and the remaining expression.

ab + ac = a(b + c)

Where:

• a is the common factor of both terms

• b + c is the remaining expression after dividing each term by a

• The common factor can be a number, a variable, or both
  
  

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• Important Questions on Factorization - Class 8

Solved Examples On Factorisation by Common Factor

Example 1: Numerical common factor

Problem: Factorise: 6x + 18

Solution:

Steps: HCF of 6 and 18 = 6

6x ÷ 6 = x

18 ÷ 6 = 3

6x + 18 = 6(x + 3)

6(x + 3) = 6x + 18.

Answer: 6x + 18 = 6(x + 3)

Example 2: Variable common factor

Problem: Factorise: x² + 5x

Solution:

Steps: Common variable: x (lowest power in both terms)

x² ÷ x = x

5x ÷ x = 5

x² + 5x = x(x + 5)

x(x + 5) = x² + 5x.

Answer: x² + 5x = x(x + 5)

Example 3: Both numerical and variable common factor

Problem: Factorise: 8x² + 12x

Solution:

Steps: HCF of 8 and 12 = 4

Common variable: x (lowest power)

Common factor = 4x

8x² ÷ 4x = 2x

12x ÷ 4x = 3

8x² + 12x = 4x(2x + 3)

4x(2x + 3) = 8x² + 12x.

Answer: 8x² + 12x = 4x(2x + 3)

Example 4: Three terms

Problem: Factorise: 15x³ + 10x² + 5x

Solution:

Steps: HCF of 15, 10, 5 = 5

Common variable: x (lowest power across all terms)

Common factor = 5x

15x³ ÷ 5x = 3x²

10x² ÷ 5x = 2x

5x ÷ 5x = 1

15x³ + 10x² + 5x = 5x(3x² + 2x + 1)

5x(3x² + 2x + 1) = 15x³ + 10x² + 5x.

Answer: 5x(3x² + 2x + 1)

Example 5: Two variables

Problem: Factorise: 6x²y + 9xy²

Solution:

Steps: HCF of 6 and 9 = 3

Common variables: x (min power 1) and y (min power 1)

Common factor = 3xy

6x²y ÷ 3xy = 2x

9xy² ÷ 3xy = 3y

6x²y + 9xy² = 3xy(2x + 3y)

3xy(2x + 3y) = 6x²y + 9xy².

Answer: 3xy(2x + 3y)