Quick Revision Notes: Class 8 Mathematics

Chapter: Factorisation

### Concept 1: Factors of Natural Numbers and Algebraic Expressions

• Factor: When a number or an algebraic expression is expressed as a product of two or more numbers/expressions, each of them is called a factor of the given number/expression.
• Irreducible Factor: A factor cannot be expressed further as a product of factors. It is like a prime number in arithmetic.
• Factorisation: The process of writing a given expression as a product of its factors (irreducible forms) is called factorisation.

### Concept 2: Methods of Factorisation

1. Method of Common Factors

Look for common numerical or literal factors in all the terms of the given expression and take them out using the distributive law: ab + ac = a(b + c).

• Example: 6xy + 9x^2 = 3x(2y + 3x)

2. Factorisation by Grouping

Sometimes, no single factor is common to all terms, but we can group the terms in pairs such that a common factor appears from each group.

• Example: ax + ay + bx + by $= a(x + y) + b(x + y)$ $= (x + y)(a + b)$

3. Factorisation using Identities

Standard algebraic identities are extremely helpful in factorisation:

• Identity I: (a + b)^2 = a^2 + 2ab + b^2 $\rightarrow$ a^2 + 2ab + b^2 = (a + b)^2
• Identity II: (a - b)^2 = a^2 - 2ab + b^2 $\rightarrow$ a^2 - 2ab + b^2 = (a - b)^2
• Identity III: (a + b)(a - b) = a^2 - b^2 $\rightarrow$ a^2 - b^2 = (a + b)(a - b)
• Identity IV: (x + a)(x + b) = x^2 + (a + b)x + ab $\rightarrow$ x^2 + (a + b)x + ab = (x + a)(x + b)

4. Factorisation of the form x^2 + px + q (Splitting the Middle Term)

To factorise expressions like x^2 + sx + p:

1. Find two factors of p whose sum is equal to s.
2. Split the middle term using these two factors.
3. Factorise by grouping.

### Concept 3: Division of Algebraic Expressions

• Division of a monomial by another monomial: Divide the numerical coefficients and subtract the powers of identical variables using the law of exponents (x^m / x^n = x^(m-n)).

  • Example: 15x^3 \div 3x = (15/3) \times (x^3 / x^1) = 5x^2

• Division of a polynomial by a monomial: Divide each term of the polynomial by the monomial separately.

  • Example: (6x^3 + 4x^2) \div 2x = (6x^3 / 2x) + (4x^2 / 2x) = 3x^2 + 2x

• Division of a polynomial by another polynomial:

  1. Factorise both the numerator and the denominator completely.
  2. Cancel out the common irreducible factors from the numerator and the denominator.
  • OR use long division method for polynomials.

### Important Tips for MP Board Exam

• Always check if a common factor can be taken out before applying identities.
• Count the number of terms: 2 terms often point to a^2 - b^2; 3 terms usually point to standard identities (a^2 \pm 2ab + b^2) or splitting the middle term.
• Always write the final answer as a product of factors.