- 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.
- 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)
- 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
- 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 18||Sum 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)
- 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)