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
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
Examples
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.
