Notes on Class 8 Maths Chapter 14: Factorisation

Introduction to Factorisation

Factorisation is a fundamental concept in algebra where an algebraic expression is expressed as a product of its factors. These factors can be numbers, variables, or algebraic expressions. For Class 8 students, this chapter introduces the process of breaking down polynomials into simpler components, which is essential for solving equations, simplifying expressions, and understanding higher-level mathematics.

Factorisation is the reverse of multiplication. For example, multiplying (2) and (3) gives (6), while factorising (6) means writing it as (2 \times 3). Similarly, in algebra, an expression like (x^2 + 5x + 6) can be factorised into ((x + 2)(x + 3)).

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Key Concepts

1. Factors and Multiples

• Factors: Numbers or expressions that divide another number or expression exactly without leaving a remainder. For example, the factors of (12) are (1, 2, 3, 4, 6, 12).

• Multiples: A multiple is the result of multiplying a number by an integer. For example, multiples of (3) are (3, 6, 9, 12), etc.

• In algebra, factors of an expression are the expressions that, when multiplied, produce the original expression.

2. Types of Expressions

• Monomials: Expressions with a single term, e.g., (5x^2).

• Binomials: Expressions with two terms, e.g., (x + 3).

• Trinomials: Expressions with three terms, e.g., (x^2 + 5x + 6).

• Polynomials: Expressions with one or more terms, e.g., (2x^3 + 3x^2 – 5x + 7).

3. Common Factors

The first step in factorisation is to identify common factors among the terms of an expression. For example:

• In (6x + 9y), the common factor is (3). Factorising gives: (3(2x + 3y)).

• In (4x^2 + 8x), the common factor is (4x). Factorising gives: (4x(x + 2)).

4. Methods 

Several methods are used to factorise algebraic expressions, depending on their structure.

a) Factorisation by Taking Out Common Factors

• Identify the greatest common factor (GCF) of all terms.

• Divide each term by the GCF and write the expression as a product.

• Example: Factorise (12x^3 + 18x^2).

  • GCF is (6x^2).

  • Divide: (\frac{12x^3}{6x^2} = 2x), (\frac{18x^2}{6x^2} = 3).

  • Result: (6x^2(2x + 3)).

b) Factorisation by Grouping

• Used when an expression has four or more terms.

• Group terms with common factors, factorise each group, and then factor out the common binomial.

• Example: Factorise (ax + ay + bx + by).

  • Group: ((ax + ay) + (bx + by)).

  • Factor each group: (a(x + y) + b(x + y)).

  • Factor out the common binomial: ((x + y)(a + b)).

c) Factorisation of Quadratic Expressions

• Quadratic expressions are of the form (ax^2 + bx + c).

• To factorise, find two numbers whose product is (ac) and sum is (b).

• Example: Factorise (x^2 + 5x + 6).

  • Find numbers whose product is (6) (constant term) and sum is (5) (coefficient of (x)).

  • Numbers are (2) and (3) ((2 \times 3 = 6), (2 + 3 = 5)).

  • Rewrite: (x^2 + 2x + 3x + 6).

  • Group: ((x^2 + 2x) + (3x + 6) = x(x + 2) + 3(x + 2)).

  • Factor: ((x + 2)(x + 3)).

d) Factorisation Using Identities

Algebraic identities simplify factorisation for specific forms of expressions. Key identities include:

• ((a + b)^2 = a^2 + 2ab + b^2)

• ((a – b)^2 = a^2 – 2ab + b^2)

• ((a + b)(a – b) = a^2 – b^2)

• Example: Factorise (x^2 + 6x + 9).

  • Recognize it as ((x + 3)^2) since (a^2 = x^2), (2ab = 6x), (b^2 = 9).

  • Result: ((x + 3)(x + 3)).

• Example: Factorise (4x^2 – 25).

  • Recognize it as (a^2 – b^2 = (2x)^2 – (5)^2).

  • Result: ((2x – 5)(2x + 5)).

5. Factorisation of Expressions with Negative or Fractional Coefficients

• Handle negative signs by factoring out (-1) if necessary.

• Example: Factorise (-x^2 – 2x + 3).

  • Factor out (-1): (-1(x^2 + 2x – 3)).

  • Factorise the quadratic: (x^2 + 2x – 3 = (x + 3)(x – 1)).

  • Result: (-1(x + 3)(x – 1)).

• For fractional coefficients, clear fractions by multiplying through.

• Example: Factorise (\frac{1}{2}x^2 + x + \frac{1}{2}).

  • Multiply by (2): (x^2 + 2x + 1).

  • Factorise: ((x + 1)^2).

  • Divide by (2): (\frac{1}{2}(x + 1)^2).

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Steps for Factorisation

1. Check for Common Factors: Always start by taking out the GCF.

2. Identify the Type of Expression:

   • Two terms: Check for difference of squares ((a^2 – b^2)).

   • Three terms: Try to factorise as a quadratic.

   • Four or more terms: Use grouping.

3. Apply Identities: If the expression matches a known identity, use it.

4. Verify: Multiply the factors to ensure the original expression is obtained.

5. Simplify: Ensure the factors are in their simplest form.

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Applications

• Simplifying Expressions: Factorisation reduces complex expressions to simpler forms.

• Solving Equations: Factorising polynomials helps find roots. For example, to solve (x^2 – 4 = 0), factorise as ((x – 2)(x + 2) = 0), so (x = 2) or (x = -2).

• Algebraic Fractions: Factorisation is used to simplify fractions like (\frac{x^2 – 1}{x – 1}).

• Geometry and Word Problems: Factorisation helps solve problems involving areas, volumes, or quantities.

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Common Mistakes to Avoid

1. Forgetting Common Factors: Always check for a GCF before proceeding.

2. Incorrect Splitting of Middle Term: Ensure the numbers chosen for quadratic factorisation satisfy both the product and sum conditions.

3. Ignoring Signs: Pay attention to positive and negative signs, especially in identities.

4. Incomplete Factorisation: Ensure the expression is fully factorised. For example, (2x^2 + 4x = 2x(x + 2)), not just (x(2x + 4)).

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Practice Examples

1. Factorise (15x^2 – 25x):

   • GCF is (5x).

   • Result: (5x(3x – 5)).

2. Factorise (x^2 – 7x + 12):

   • Find numbers: Product = (12), Sum = (-7).

   • Numbers are (-3) and (-4).

   • Result: ((x – 3)(x – 4)).

3. Factorise (9x^2 – 16y^2):

   • Use (a^2 – b^2): ((3x)^2 – (4y)^2).

   • Result: ((3x – 4y)(3x + 4y)).

4. Factorise (2x + 2y + ax + ay):

   • Group: ((2x + 2y) + (ax + ay)).

   • Factor: (2(x + y) + a(x + y)).

   • Result: ((x + y)(2 + a)).

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Tips for Mastery

• Practice identifying patterns in expressions to apply identities quickly.

• Verify factorisation by expanding the factors.

• Solve a variety of problems to understand different types of expressions.

• Use factorisation to simplify real-life problems, such as dividing quantities or solving geometric puzzles.

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Conclusion

Factorisation is a powerful tool in algebra that simplifies expressions and solves equations. By mastering common factors, grouping, quadratic factorisation, and identities, students can tackle a wide range of problems. Regular practice and attention to detail will build confidence and proficiency in this essential mathematical skill.

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