Factorisation Method: Definition, Properties, Steps, and Solved Examples

Factorization is a fundamental mathematical technique that simplifies complex expressions and equations by breaking them down into simpler, more manageable components. This process, often referred to as the reverse of multiplication, involves expressing a number or algebraic expression as a product of its factors. In this blog, we will delve into the factorization method, exploring its definition and the important properties of the factorisation method. We will also outline the step-by-step approach to factorizing expressions and provide solved examples to solidify understanding. Key terms, such as factors, prime numbers, and polynomial equations, will be defined and contextualized throughout the discussion, offering a comprehensive guide to mastering factorization. Whether you are a student seeking to enhance your mathematical skills or a professional looking to refresh your knowledge, this blog will serve as a valuable resource on the path to factorization proficiency.

Definition of Factorisation

Factorisation is the mathematical process of breaking down a number, polynomial, or algebraic expression into a product of simpler elements called factors, which, when multiplied together, yield the original number or expression. In other words, it is the reverse process of multiplication, where the goal is to identify the basic components or building blocks that, when combined, form the given entity. For example, the number 12 can be factorised into 2, 2, and 3 (i.e., 12 = 2×2×3), and the polynomial x²−4 can be factorised into (x−2)(x+2).

Important Properties of Factorization:

1. Uniqueness (Fundamental Theorem of Arithmetic): Every integer greater than 1 can be represented uniquely as a product of prime factors, up to the order of the factors.
2. Simplicity: Factoring simplifies complex expressions, making them easier to solve or manipulate in equations.
3. Distributive Property: This property is often used in factorization, allowing the expression of a sum as a product. For example, ab+ac = a(b+c).
4. Zero Product Property: If a product of factors equals zero, then at least one of the factors must be zero. This property is crucial in solving polynomial equations.
5. Commutative and Associative Properties: These properties facilitate the rearrangement and grouping of factors during factorization.

Also Read: Variables and Constants

Properties of Factorisation Method

Here are the properties of the factorization method with examples for clarity:

1. Uniqueness (Fundamental Theorem of Arithmetic): Every integer greater than 1 has a unique prime factorization. For example, the number 60 can be factored into prime numbers as follows:
   60 = 2²×3×5.
   This is the only way to represent 60 as a product of primes, disregarding the order of the factors.
2. Commutative Property: The order of the factors does not affect the product. For example:
   3×5 = 5×3
   Both expressions yield the same product, which is 15.
3. Associative Property: The way in which factors are grouped does not change the product. For example:
   (2×3)×4 = 2×(3×4)
   Both groupings result in a product of 24.
4. Distributive Property: A common factor can be distributed across terms within an expression. For instance:
   3x+3y = 3(x+y)
   Here, the factor 3 is distributed across both x and y.
5. Zero Product Property: If the product of several factors is zero, at least one of the factors must be zero. For example, in the equation:
   (x−2)(x+3) = 0
   Either x−2 = 0 or x+3 = 0. Thus, the solutions are x = 2 or x = −3.
6. Factorization of Polynomials: Polynomials can be factored into simpler polynomials or monomials. For example:
   x²−5x+6 can be factored as (x−2)(x−3).
   This factorization reveals the roots of the polynomial, which are x=2 and x=3.
7. Simplification: Factorization simplifies expressions, making them easier to work with. For example, simplifying 12x²y+18xy² involves factoring out the greatest common factor:
   6xy(2x+3y).
8. Applicability to Various Forms: Factorization applies to numbers, algebraic expressions, and matrices. For instance, factoring the matrix [4,2 2,1] could involve finding scalar multiples or identifying a simpler matrix structure.

Steps to Solve Factorisation Method

Here are the steps to solve factorization problems, explained with examples:

1. Identify the Expression to be Factorized

Example: Factorize x²+5x+6.

2. Look for Common Factors

Example: For 6x²+9x, factor out the greatest common factor (GCF):

6x²+9x = 3x(2x+3)

3. Factorize Completely

a. For Quadratic Polynomials: Find two numbers that multiply to give the constant term (in this case, 6) and add to give the coefficient of the middle term (5).

Example: For x²+5x+6, the numbers are 2 and 3 because 2×3=6 and 2+3 = 5.

Thus, the factorization is:

x²+5x+6 = (x+2)(x+3)

b. For Difference of Squares:

• Use the formula a²−b² = (a−b)(a+b).

Example: For x²−9:

x²−9 = (x−3)(x+3)

c. For Perfect Square Trinomials:

• Use the formula a² ± 2ab + b² = (a±b)².

Example: For x²+6x+9:

x²+6x+9 = (x+3)²

d. For Polynomials by Grouping: Group terms and factor each group separately, then factor out the common binomial factor.

Example: For x³+3x²+2x+6:

x³+3x²+2x+6 = (x³+3x²) + (2x+6)

= x²(x + 3) + 2(x + 3)

= (x² + 2)(x + 3)

4. Recheck

Example: Verify the factorization of x²+5x+6 by expanding (x+2)(x+3):

(x+2)(x+3) = x²+3x+2x+6

= x²+5x+6

The original expression and the expanded result match, confirming the factorization is correct.

5. Simplify if Needed

Example: For the expression (6x²+9x)/3x​, simplify by factoring:

= (6x²+9x)/3x 

= {3x(2x+3)}/3x

= 2x+3

Also Read: Heights and Distances

Factorisation Method Solved Examples

Here are five solved examples of the factorization method:

Example 1: Factoring a Quadratic Expression

Expression: x²+9x+20

Steps:

1. Find two numbers that multiply to 20 (constant term) and add to 9 (coefficient of x). These numbers are 4 and 5.
2. Write the factorization: x²+9x+20 = (x+4)(x+5)

Verification:

= (x+4)(x+5) 

= x²+5x+4x+20 

= x²+9x+20

Example 2: Factoring a Difference of Squares

Expression: 49x²−36

Steps:

1. Recognize this as a difference of squares: a²−b² = (a−b)(a+b).
2. Rewrite 49x² as (7x)² and 36 as 6².
3. Apply the formula: 49x²−36 = (7x)²−² = (7x−6)(7x+6)

Verification:

(7x−6)(7x+6) = 49x²+42x−42x−36 = 49x²−36

Example 3: Factoring a Perfect Square Trinomial

Expression: x²+10x+25

Steps:

1. Recognize this as a perfect square trinomial: a²+2ab+b² = (a+b)².
2. Here, x² is a², 25 is b², and 10x is 2ab.
3. Apply the formula: x²+10x+25 = (x+5)²

Verification:

(x+5)² = x²+10x+25

Example 4: Factoring by Grouping

Expression: x³+4x²−3x−12

Steps:

1. Group terms: (x³+4x²)−(3x+12).
2. Factor out the common factors: x²(x+4)−3(x+4)
3. Factor out the common binomial factor: (x²−3)(x+4)

Verification:

(x²−3)(x+4) = x³+4x²−3x−12

Example 5: Factoring a Polynomial with a Common Factor

Expression: 15x³−10x²+5x

Steps:

1. Find the greatest common factor (GCF): The GCF of 15x³, −10x², and 5x is 5x.
2. Factor out the GCF: 15x³−10x²+5x = 5x(3x²−2x+1)

Verification:

5x(3x²−2x+1) = 15x³−10x²+5x

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