Chapter 2: Factorisation and Algebraic Fractions

2.1 Expansion of Algebraic Expressions

Expansion is the product of a term or algebraic expression with another algebraic expression, effectively removing the brackets.

Single Bracket Expansion

To expand expressions with one bracket, multiply the outer term by each term inside the bracket:

• $a(b + c) = ab + ac$
• $a(b - c) = ab - ac$
• $-a(b + c) = -ab - ac$

Two Brackets Expansion

To expand two binomial expressions, multiply every term in the first bracket by every term in the second bracket using the distributive property (FOIL method: First, Outside, Inside, Last):

$$(a + b)(c + d) = a(c + d) + b(c + d) = ac + ad + bc + bd$$

Special Algebraic Identities

Memorising these special expansion patterns simplifies working with squared binomials:

• $(a + b)^2 = a^2 + 2ab + b^2$
• $(a - b)^2 = a^2 - 2ab + b^2$
• $(a + b)(a - b) = a^2 - b^2$

2.2 Factorisation of Algebraic Expressions

Factorisation is the process of writing an algebraic expression as a product of its factors. It is the reverse process of expansion.

Methods of Factorisation

1. Highest Common Factor (HCF): Extract the HCF of the terms out of the bracket.

   Example: $6x + 9 = 3(2x + 3)$

2. Difference of Two Squares ($a^2 - b^2$):

   Formula: $a^2 - b^2 = (a + b)(a - b)$

   Example: $x^2 - 16 = x^2 - 4^2 = (x + 4)(x - 4)$

3. Cross Multiplication Method ($ax^2 + bx + c$):

   Used to factorise quadratic trinomials into two linear factors $(px + q)(rx + s)$.

   Example: $x^2 + 5x + 6 = (x + 2)(x + 3)$

4. Grouping Method (4 Terms):

   Group terms in pairs with common factors and factorise each pair step-by-step.

   Formula: $ab + ac + bd + cd = a(b + c) + d(b + c) = (a + d)(b + c)$

2.3 Algebraic Expressions and Basic Operations

An algebraic fraction is a fraction where the numerator, denominator, or both are algebraic expressions.

Addition and Subtraction of Algebraic Fractions

• Same Denominator: Combine the numerators directly over the common denominator. $$\frac{a}{c} + \frac{b}{c} = \frac{a + b}{c}$$
• Different Denominators: Find the Lowest Common Multiple (LCM) of the denominators before adding or subtracting. $$\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}$$

Multiplication and Division of Algebraic Fractions

• Multiplication: Factorise numerators and denominators first, then cancel common factors before multiplying directly. $$\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}$$
• Division: Multiply by the reciprocal of the second fraction and simplify. $$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{ad}{bc}$$