Expanding and Factorising Brackets - 10/6/25

Expanding Single Brackets

  1. Basic principle: Multiply each term inside the bracket by the term outside the bracket.

  2. Example:
    a(b + c) = a \cdot b + a \cdot c = ab + ac

  3. Numerical Example:
    3(x + 2) = 3 \cdot x + 3 \cdot 2 = 3x + 6

  4. Example with a negative term:
    5(x - 3) = 5 \cdot x - 5 \cdot 3 = 5x - 15

  5. Example with a variable outside:
    x(2x + 4) = x \cdot 2x + x \cdot 4 = 2x^2 + 4x

  6. Example with negative variable outside: -2x(x - 5) = -2x \cdot x - (-2x) \cdot 5 = -2x^2 + 10x

    • Note the sign change when multiplying negative terms.

Expanding Double Brackets

  1. FOIL Method: A common mnemonic to remember the steps:

    • First: Multiply the first terms in each bracket.

    • Outer: Multiply the outer terms in each bracket.

    • Inner: Multiply the inner terms in each bracket.

    • Last: Multiply the last terms in each bracket.

  2. General Form:
    (a+b)(c+d) = a \cdot c + a \cdot d + b \cdot c + b \cdot d

  3. Example:
    (x + 2)(x + 3) = x \cdot x + x \cdot 3 + 2 \cdot x + 2 \cdot 3 = x^2 + 3x + 2x + 6 = x^2 + 5x + 6

  4. Example with negative terms:
    (x - 4)(x + 2) = x \cdot x + x \cdot 2 - 4 \cdot x - 4 \cdot 2 = x^2 + 2x - 4x - 8 = x^2 - 2x - 8

  5. Example with more complex terms:
    (2x + 1)(x - 3) = 2x \cdot x + 2x \cdot (-3) + 1 \cdot x + 1 \cdot (-3) = 2x^2 - 6x + x - 3 = 2x^2 - 5x - 3

Definition:
Factorising is the process of breaking down an expression into a product of simpler expressions (called factors) that, when multiplied together, give the original expression.

Why Factorise?

  • To simplify expressions

  • To solve equations (by setting factors to zero)

  • To find common factors in expressions

  • To make expressions easier to work with in algebraic manipulations

Common Types of Factorising

  1. Factorising out the Greatest Common Factor (GCF)
    Example:
    6x³ + 9x² = 3x² (2x + 3)
    Here, 3x² is the GCF.

  2. Factorising Quadratics (Trinomials)
    For quadratic expressions of the form ax² + bx+ c , factor into two binomials:
    Example:
    x² + 5x +6 = (x + 2)(x + 3)

  3. Difference of Squares
    a² − b² = (a - b)(a + b)

    Example:
    x² − 16 = (x - 4)(x + 4)

  4. Perfect Square Trinomials
    a² + 2ab + b² = (a + b)²

    a² − 2ab + b² = (a - b)²
    Example:
    x² + 6x + 9 = (x + 3)²

Steps for Factorising Quadratics (when a = 1):

  1. Find two numbers that multiply to c (constant term) and add to b (middle term coefficient).

  2. Rewrite the middle term using these two numbers.

  3. Factor by grouping or write directly as binomials.