# Chapter 12: Factorisation

## Overview
Factorisation is the process of writing an algebraic expression as a product of its factors. Just as natural numbers can be expressed as products of their prime factors, algebraic expressions can be factorised into their irreducible components. This chapter introduces systematic methods for factorisation, which is essential for simplifying algebraic expressions and solving equations.

## Key Concepts

### 12.1 Introduction to Factorisation

#### Factors of Natural Numbers
- **Definition**: A factor of a number is a value that divides the number exactly
- **Prime Factorisation**: Writing a number as a product of prime factors
- **Example**: 30 = 2 × 3 × 5 (prime factor form)

#### Factors of Algebraic Expressions
- **Irreducible Factors**: Factors that cannot be expressed further as products
- **Example**: In 5xy, the irreducible factors are 5, x, and y
- **Note**: 1 is a factor of every term but is usually not shown separately

### 12.2 Methods of Factorisation

#### Method 1: Common Factor Method
The most basic method involves finding factors common to all terms.

**Steps**:
1. Write each term as a product of irreducible factors
2. Identify common factors
3. Use distributive law to factor out common elements

**Example**:
- 2x + 4 = 2(x + 2)
- 5xy + 10x = 5x(y + 2)

#### Method 2: Factorisation by Regrouping
When terms don't have a single common factor, group terms strategically.

**Process**:
- Group terms with common factors
- Factor each group separately
- Look for common factors across groups

**Example**:
2xy + 2y + 3x + 3 = 2y(x + 1) + 3(x + 1) = (x + 1)(2y + 3)

#### Method 3: Using Algebraic Identities
Apply standard algebraic identities for specific patterns.

**Key Identities**:
- (a + b)² = a² + 2ab + b² (Perfect Square Trinomial)
- (a - b)² = a² - 2ab + b² (Perfect Square Trinomial)
- (a + b)(a - b) = a² - b² (Difference of Squares)
- (x + a)(x + b) = x² + (a + b)x + ab

**Examples**:
- x² + 8x + 16 = (x + 4)²
- 49p² - 36 = (7p)² - 6² = (7p - 6)(7p + 6)

#### Method 4: Factors of the Form (x + a)(x + b)
For quadratic expressions x² + px + q:
- Find two numbers a and b such that ab = q and a + b = p
- Factor as (x + a)(x + b)

**Example**: x² + 5x + 6
- Need ab = 6 and a + b = 5
- Try a = 2, b = 3: 2 × 3 = 6 ✓ and 2 + 3 = 5 ✓
- Therefore: x² + 5x + 6 = (x + 2)(x + 3)

### 12.3 Division of Algebraic Expressions

#### Division of Monomial by Monomial
Cancel common factors in numerator and denominator.

**Example**: 6x³ ÷ 2x = (2 × 3 × x × x × x) ÷ (2 × x) = 3x²

#### Division of Polynomial by Monomial
Divide each term of the polynomial by the monomial.

**Example**: (4y³ + 5y² + 6y) ÷ 2y = 2y² + (5/2)y + 3

#### Division of Polynomial by Polynomial
Factor both expressions and cancel common factors.

**Example**: (7x² + 14x) ÷ (x + 2) = 7x(x + 2) ÷ (x + 2) = 7x

## Real-World Applications

### 1. Engineering and Construction
- **Structural Analysis**: Factorising expressions for load calculations
- **Material Optimisation**: Simplifying area and volume formulas
- **Circuit Design**: Factorising electrical resistance expressions

### 2. Computer Science
- **Algorithm Optimisation**: Factorising complexity expressions
- **Graphics Programming**: Simplifying coordinate transformation formulas
- **Data Compression**: Using factorisation for efficient storage

### 3. Physics and Chemistry
- **Kinematic Equations**: Factorising motion-related expressions
- **Chemical Reactions**: Balancing equations through factorisation
- **Wave Analysis**: Simplifying frequency and amplitude expressions

### 4. Economics and Finance
- **Cost Analysis**: Factorising profit and loss expressions
- **Investment Calculations**: Simplifying compound interest formulas
- **Market Modelling**: Factorising supply and demand equations

### 5. Architecture and Design
- **Area Calculations**: Factorising complex geometric expressions
- **Pattern Design**: Using factorisation for symmetrical layouts
- **Scale Modelling**: Simplifying proportion calculations

## Problem-Solving Strategies

### Strategy 1: Identify the Pattern
- Look for perfect squares, differences of squares, or common factors
- Check if the expression fits standard identity forms
- Examine coefficients and constant terms for factorisation clues

### Strategy 2: Choose the Right Method
- **Common factors first**: Always check for common factors before other methods
- **Regrouping**: Use when no single common factor exists
- **Identities**: Apply when expressions match standard forms
- **Trial and error**: For quadratic expressions, systematically try factor pairs

### Strategy 3: Verification
- Always multiply factors to verify the original expression
- Check that all factors are in irreducible form
- Ensure no common factors remain unfactored

## Study Tips

### Mastering Identities
1. **Memorise Key Patterns**: Learn to recognise a² + 2ab + b², a² - b², etc.
2. **Practice Recognition**: Quickly identify which identity applies
3. **Backwards Thinking**: Start with factored form and expand to understand patterns

### Common Mistakes to Avoid
1. **Incomplete Factorisation**: Always factor completely
2. **Sign Errors**: Pay careful attention to positive and negative signs
3. **Missing Common Factors**: Always check for common factors first
4. **Verification**: Always multiply back to check your answer

### Systematic Approach
1. Look for common factors across all terms
2. If no common factor, try regrouping
3. Check if any identities apply
4. For quadratics, use the (x + a)(x + b) method
5. Verify by multiplying factors

## Advanced Concepts

### Multiple Factorisation Steps
Some expressions require applying multiple methods:
- Factor out common factors first
- Then apply identities or other methods
- Continue until all factors are irreducible

### Higher Degree Polynomials
- May require multiple applications of identities
- Look for patterns like a⁴ - b⁴ = (a² - b²)(a² + b²)
- Consider substitution methods for complex expressions

### Connection to Equation Solving
- Factorisation is essential for solving quadratic equations
- Zero Product Property: If ab = 0, then a = 0 or b = 0
- Factorised form reveals roots directly

This chapter provides the foundation for advanced algebraic manipulation and is crucial for success in higher mathematics, including calculus and advanced algebra.