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Factorization

  • To factorize an expression is to write it as a product of its factors.
  • In algebra, factorization is the opposite of removing brackets.
  • If the terms in an algebraic expression have common factors, the expression can be written as a product that includes the common factor.
  • Larger algebraic expressions can sometimes be factorized by grouping.
Examples
  1. Factorize cx + cy + 2dx + 2dy
    • The terms cx and cy have c in common.
    • The terms 2dx and 2dy have 2d in common.
    • By grouping the expression into pairs: (cx + cy) + (2dx + 2dy)
    • ⇒ c(x + y) + 2d(x + y)
    • The two products now have (x + y) in common
    • ⇒ (c + 2d) (x + y)
    • Hence cx + cy + 2dx + 2dy = (c + 2d) (x + y)
  2. Factorize 2sru + 6tru – 4srv – 12trv
    • 2r is a factor of every term in the given expression.
    • ⇒ 2r[su + 3tu – 2sv – 6tv]
    • ⇒ 2r[u(s + 3t) – 2v(s + 3t)]
    • ⇒ 2r(s + 3t)(u – 2v)
FACTORIZATION OF QUADRATIC EXPRESSIONS
  • To expand an expression in the form (a + b)(c + d), means to find the product that results from multiplying each term in the first binomial bracket by each item in the second binomial bracket.
  • A quadratic expression is one in which 2 is the highest power of the unknown.
    For example: y² – 3y + 2 is a quadratic expression.
  • Remember the following identities for perfect squares:
    • (a + b)² = a² + 2ab + b², and
    • (a – b)² = a² – 2ab + b²
  • The difference of two squares is factorized as follows:
    • a² – b² = (a + b)(a – b)
    • NOTE: (a – b)² ≠ a² – b². For example:
      • Let a = 9, and b = 3.
      • (a – b)² = (9 – 3)² = (6)² = 36
      • a² – b² = 9² – 3² = 81 – 9 = 72
      • 36 ≠ 72. ∴ (a – b)² ≠ a² – b²
Examples
  1. Factorize d² + 11d + 18
    • 1st Step: d² + 11d + 18 = (d     )(d      )
    • 2nd Step: Find two numbers such that their product is 18 and their sum is +11. Since the 18 and 11 are positive, consider positive factors only.
      Factors of 18Sum of Factors
      a.+1 and +18+19
      b.+2 and +9+11
      c.+3 and +6+9
    • Of these, only b gives the required result.
    • Thus, d² + 11d + 18 = (d + 2)(d + 9)
  2. Factorize 5a² – 45.
    • The two terms have the factor 5 in common, take this out first.
    • ⇒ 5a² – 45 = 5(a² – 9)
    • = 5(a² – 3²)
    • = 5(a + 3)(a – 3)
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