Class VIII Math
Notes for Factorisation
   Factorisation means write an expression as a product of' its factors.
   Like prime factors, an irreducible factor, a factor which cannot be expressed further as a product of factors.
   Some expression can easily be factorised using these identities:
      I. a2 + 2ab + b2 = (a + b)2
      II. a2 – 2ab + b2 = (a – b)2
      III. a2 – b2 = (a – b)(a + b)
      IV. x2 + (a + b)x + ab = (x + a)(x + b)
   The number 1 is a factor of every algebraic term also, but it is shown only when needed.
   When factorisation of x2 + (a + b)x + ab is done by splitting the middle term, the two numbers which give the product ab and (a + b) as the coefficient of x have to be chosen very caully with correct sign.
      (i) a2 + 2ab + b2 = (a + b)2
      (ii) a2 – 2ab + b2 = (a – b)2
      (iii) a2 – b2 = (a – b)(a + b)
      (iv) 1 is a factor of every term of an algebraic expressionless it is specially required, we do not show I as a separate factor of any term.
      (v) Factorisation means writing an expression as product Of factors.
Note: In case of factorisation of a term of an expression, the word 'irreducible' is used in place of 'prime'.
      For example, 6pq = 2 × 3 × pq is not the irreducible corm because pq can further be factorised as p × q, i.e. the irreducible fonn of 6pq = 2 × 3 × p × q.
Example: Write 10xy as irreducible factor form.
Solution: We have          10 = 2 × 5
                                                   xy = x × y
                                                   10xy = 2 × 5 × x × y