Factorisation

• What is Factorisation?

  • Definition of Factors of a natural number
  • Definition of Factors of an algebraic expression
  • Meaning of Irreducible factor
  • Factorisation as the reverse process of Multiplication

• Methods of Factorisation

  • Method of Common Factors
    • Find common factors of each term
    • Distributive property: $ab + ac = a(b + c)$
    • Example: $2x + 4 = 2(x + 2)$
  • Factorisation by Regrouping Terms
    • Group terms with a common factor
    • Factor out the common binomial
    • Example: $2x + 2y + cx + cy = 2(x + y) + c(x + y) = (2 + c)(x + y)$
  • Factorisation using Identities
    • Identity 1: $(a + b)^2 = a^2 + 2ab + b^2$
    • Identity 2: $(a - b)^2 = a^2 - 2ab + b^2$
    • Identity 3: $a^2 - b^2 = (a + b)(a - b)$
    • Identity 4: $(x + a)(x + b) = x^2 + (a + b)x + ab$
  • Factors of the form $(x + a)(x + b)$
    • Splitting the middle term
    • Example: $x^2 + 5x + 6 \rightarrow x^2 + 2x + 3x + 6 = (x + 2)(x + 3)$

• Division of Algebraic Expressions

  • Division of a monomial by another monomial
    • Example: $7x^2 \div 14x$
  • Division of a polynomial by a monomial
    • Divide each term of the polynomial by the monomial
    • Example: $(5x^2 - 6x) \div x$
  • Division of a polynomial by a polynomial
    • Factorise both numerator and denominator first
    • Cancel common irreducible factors
    • Example: $(y^2 + 7y + 10) \div (y + 5)$

• Errors to be Avoided

  • Misinterpreting brackets in negative signs: $-(x - 4) = -x + 4$ (not $-x - 4$)
  • Squaring a binomial: $(x + 2)^2 = x^2 + 4x + 4$ (not $x^2 + 4$)
  • Correct substitution of variables in negative values: $(-3)^2 = +9$ (not $-9$)
  • Distributing division correctly over all terms in the numerator