Introduction
Factorisation is the process of breaking down an algebraic expression or a number into its factors. Factors are numbers or expressions that, when multiplied together, give the original number or expression.
For example, 12 can be factorised as 2 × 2 × 3 or 4 × 3. Similarly, x² - 9 can be factorised as (x - 3)(x + 3).
Types of Factorisation
- Factorisation by Taking Common Factors
- Factorisation by Grouping Terms
- Factorisation using Identities
- Factorisation of Quadratic Expressions
- Factorisation of Cubic Expressions
- Factorisation by Division Method
Factorisation by Taking Common Factors
Example: Factorise 6x² + 9x.
Solution: 6x² + 9x = 3x(2x + 3)
Factorisation by Grouping Terms
Example: Factorise ax + ay + bx + by.
Solution: (ax + ay) + (bx + by) = a(x + y) + b(x + y) = (a + b)(x + y)
Factorisation using Identities
Example: Factorise x² - 16.
Solution: x² - 16 = (x - 4)(x + 4)
Factorisation of Quadratic Expressions
Example: Factorise x² + 5x + 6.
Solution: x² + 5x + 6 = (x + 2)(x + 3)
Factorisation of Cubic Expressions
Example: Factorise x³ - 3x² - x + 3.
Solution: (x³ - 3x²) - (x - 3) = x²(x - 3) - 1(x - 3) = (x - 3)(x² - 1) = (x - 3)(x - 1)(x + 1)
Factorisation by Division Method
Example: Factorise x³ + 4x² - 3x - 12 given that x + 3 is a factor.
Solution: x³ + 4x² - 3x - 12 = (x + 3)(x + 2)(x - 2)
Applications of Factorisation
- Solving Algebraic Equations
- Finding HCF and LCM
- Geometry & Physics Applications
Conclusion
Factorisation is an essential algebraic skill that simplifies expressions and helps solve equations efficiently. By mastering different methods, we can tackle complex problems with ease.