Factorisation

In algebra, factorization is an important tool for simplifying expressions and solving equations. The process of factorization involves expressing a number, algebraic expression, or polynomial as a product of its factors. This technique is key for solving algebraic equations, simplifying expressions, and understanding the properties of numbers and variables.

This topic discusses what factorization is, different methods of factorization, important factorization formulas, and a step-by-step guide on how to factor expressions. It will also cover specific techniques like the common factors method, regrouping terms method, factorization using identities, and the factorization of quadratic polynomials. All the content includes solved examples and practice problems to help students understand and apply the concepts easily.

 

Table of Contents

• What is Factorisation
• Factorisation in Algebra
• Factorisation Formulas
• How to Factorise
  • Common Factors Method
  • Regrouping Terms Method
  • Factorisation Using Identities
  • (x + a)(x + b) Form
• Factorisation of Quadratic Polynomial
• Solved Examples
• Practice Questions
• Conclusion
• FAQs On Factorisation

 

What is Factorisation

Factorisation is describing a number or algebraic expression in terms of its factors. In algebra, it is dividing an expression into terms that are easier to work with and can be multiplied together to produce the original expression.

Example:

 x² - 9 = (x - 3)(x + 3)

Knowing what is factorisation assists us in simplifying expressions and solving equations.

 

Factorisation in Algebra

Factorisation in algebra refers to expressing algebraic expressions such as monomials, binomials, or polynomials as a product of two or more simpler expressions.

Factorisation is applied in solving algebraic equations and in simplifying expressions. Most algebra problems are factorisation problems.

For instance:

 x² + 5x + 6 = (x + 2)(x + 3)

Both the factorization technique and proper factorization formula for each kind of expression should be known to the students.

 

Factorisation Formulas

Here are the most frequently applied factorisation formulas:

• a² - b² = (a - b)(a + b)

• a² + 2ab + b² = (a + b)²

• a² - 2ab + b² = (a - b)²

• x² + (a + b)x + ab = (x + a)(x + b)

These formulas of factorization assist in solving quickly most factorisation problems.

 

How to Factorise

To answer factorisation questions, we apply various factorization techniques based on the type of expression.

 

Common Factors Technique

This technique consists of extracting the highest common factor (HCF) of all the terms of the given algebraic expression.

Apply this technique when every term in the expression has a common factor (either an integer or a variable, or both).

Steps:

1. Identify the common factor in all the terms.

2. Factor it out.

3. Express the given expression as a product.

 

Example 1:

Factorise: 6x + 9

Solution:

Step 1: Common factor in 6x and 9 is 3

Step 2: Factor out 3

Answer: 3(2x + 3)

 

Example 2:

Factorise: 12x²y + 8xy²

Solution:

Common factor = 4xy

Answer: 4xy(3x + 2y)

 

Regrouping Terms Method

In this approach, the given expression is rearranged (grouped) into two or more groups in such a way that each group shares a common factor. Afterward, factorisation is carried out by factoring each group and determining a common binomial.

Use it when the given expression contains four or more terms, and the direct common factor is not clear.

Steps:

1. Group the terms in pairs or appropriate parts.

2. Factor each group.

3. If both groups share a common binomial, factor it.

 

Example

Factorise: ab + a + b + 1

Solution:

Step 1: Group as (ab + a) + (b + 1)

Step 2: Take common factor from each group

 ab + a = a(b + 1)

 b + 1 = 1(b + 1)

Step 3: Now (b + 1) is common

 Answer: (a + 1)(b + 1)

 

Example:

Factorise: 2xy + 3x + 2y + 3

Solution

Group as (2xy + 3x) + (2y + 3)

= x(2y + 3) + 1(2y + 3)

= (x + 1)(2y + 3)

 

Factorisation Using Identities

Make use of algebraic identities to directly factorise expressions that are in standard forms.

When the expression is in the form of:

• a² − b²

• a² + 2ab + b²

• a² − 2ab + b²

• a³ ± b³

 

Common Identities:

• a² − b² = (a − b)(a + b)

• a² + 2ab + b² = (a + b)²

• a² − 2ab + b² = (a − b)²

• a³ + b³ = (a + b)(a² − ab + b²)

• a³ − b³ = (a − b)(a² + ab + b²)

 

Example 1:

Factorise: x² − 25

 This is in the form a² − b²

 x² − 5² = (x − 5)(x + 5)

 

Example 2:

Factorise: 4x² − 9

 = (2x)² − 3² = (2x − 3)(2x + 3)

 

Example 3:

Factorise: a³ − 8

= a³ − 2³ = (a − 2)(a² + 2a + 4)

 

Factors of the Form (x + a)(x + b)

(This is also referred to as middle-term splitting method)

It is a technique to factor quadratic trinomials (expressions of the type x² + bx + c).

Use this when:

• The expression contains three terms: x² + bx + c

• You can have two numbers a and b such that:

• a + b = middle term coefficient

• a × b = constant term

 

Steps:

1. Get two numbers a and b such that:

2. a + b = middle term coefficient

3. ab = constant term

4. Separate the middle term by a and b.

5. Group and factor.

 

Example 1:

Factorise: x² + 5x + 6

Get two numbers whose sum is 5 and product is 6

2 + 3 = 5 and 2×3 = 6

Step-by-step:

x² + 2x + 3x + 6

= x(x + 2) + 3(x + 2)

= (x + 3)(x + 2)

 

Example 2:

Factorise: x² − x − 6

Numbers that add to −1 and multiply to −6: −3 and 2

 x² − 3x + 2x − 6

= x(x − 3) + 2(x − 3)

= (x − 3)(x + 2)

 

Factorisation of Quadratic Polynomial

Factorisation of Quadratic Polynomial is writing a polynomial of the type ax² + bx + c in the form of two binomial factors.

Steps to solve:

1. Multiply a and c

2. Find two numbers with product ac and sum b

3. Split the middle term

4. Group and factor

 

Example:

 Factorise 3x² + 14x + 8

 a = 3, c = 8, so ac = 24

Numbers: 12 and 2 (as 12 × 2 = 24 and 12 + 2 = 14)

Rewrite: 3x² + 12x + 2x + 8

Group: (3x² + 12x) + (2x + 8) = 3x(x + 4) + 2(x + 4)

Final answer: (3x + 2)(x + 4)

 

This is the most widely used technique employed for factorisation of quadratic polynomial problems.

 

Solved Examples

Problem:
Factorise: 6x² + 9x

Solution:
Step 1: Find the HCF of the terms.
HCF of 6x² and 9x = 3x

Step 2: Factor out the HCF.
6x² + 9x = 3x(2x + 3)

Answer: 3x(2x + 3)

 

Problem:
Factorise: x² + 5x + 6

Solution:
Step 1: Split the middle term (5x) into two numbers whose sum is 5 and product is 6.
x² + 2x + 3x + 6

Step 2: Group and factorise each pair.
= x(x + 2) + 3(x + 2)

Step 3: Factor the common binomial.
= (x + 2)(x + 3)

Answer: (x + 2)(x + 3)

 

Problem:
Factorise: a² - 4b²

Solution:
This is a difference of squares.
Use the identity: a² - b² = (a - b)(a + b)

Here,
a² - 4b² = (a - 2b)(a + 2b)

Answer:(a - 2b)(a + 2b)

 

Problem:
Factorise:
x² - 7x + 10

Solution:
We need two numbers whose product is 10 and sum is -7.
Those numbers are -5 and -2.

x² - 7x + 10 = x² - 5x - 2x + 10

Group terms:
= x(x - 5) - 2(x - 5)

Take common:
= (x - 2)(x - 5)

Answer: (x - 2)(x - 5)

 

Problem:
Factorise: x² + 3x + 2

Solution:
We find two numbers whose sum = 3 and product = 2
That is: 1 and 2

x² + 3x + 2 = x² + x + 2x + 2

Group:
= x(x + 1) + 2(x + 1)

Factor common:
= (x + 2)(x + 1)

Answer: (x + 2)(x + 1)

These are some examples of using factorization formula and factorization method in real problems.

 

Practice Questions

1. Factorise x² + 6x + 9

2. Factorise 9a² - 25b²

3. Factorise 6x² + 13x + 6

4. Factorise 8x² - 10x - 3

5. Factorise x² + 12x + 35

6. Factorise x² - 2x - 15

7. Factorise a² + 7a + 10

8. Factorise using regrouping: ab + a + b + 1

9. Factorise x² - 81

10. Factorise 5x² + 11x + 2

 

Conclusion

Factorisation is the breaking down of algebraic expressions into simpler form or products of expressions. What is factorisation? It is a fundamental concept of algebra that simplifies and helps in solving equations. There are a number of factorization techniques such as common factors, regrouping, identities, and trinomials. Applying the right factorization formula assists you in doing factorisation problems correctly and efficiently. Practice in factorisation of quadratic polynomial is critical in advanced classes.

 

Related Links

Types of Fractions - Learn about the different types of fractions such as proper, improper, and mixed fractions, with clear definitions and examples.

Fractions - Understand the concept of fractions, their components, and how they are used in mathematics through detailed explanations and illustrations.

Addition of Fractions - Master how to add fractions with like and unlike denominators using step-by-step methods and solved examples.

Frequently Asked Questions on Factorisation

1. How to factorize step by step?

Ans: To factorize step by step, follow these methods:

• • Identify common factors in all terms.

  • Use formulas like a² - b² = (a - b)(a + b).

  • Apply regrouping or splitting the middle term in quadratic expressions.

Check if the expression can be written as a product of binomials.

2. What is 36 factorization?

Ans: The factorization of 36 involves breaking it down into prime numbers.
36 = 2 × 2 × 3 × 3 = 2² × 3²

3. What do you mean by factorisation?

Ans: Factorisation is the process of writing a number or algebraic expression as a product of its factors. In algebra, it means expressing a polynomial as a product of simpler polynomials.

4. What are the 4 types of factorisation?

Ans: The 4 main types of factorisation in algebra are:

1. Common factor method
2. Difference of squares
3. Trinomial method (splitting the middle term)
4. Grouping method
   
   

Strengthen your understanding of algebra by learning factorisation techniques through step-by-step explanations and solved examples at Orchids The International School.