Factorization
In arithmetic, some positive integers
can be written as a product of their prime factors.
For example, 600 = 23 × 3 ×
52
In algebra, some expressions can also
be expressed as a product of simpler expressions.
For example, ax + ay = a(x + y)
Here, a and (x + y) are called factors
of the given expression ax + ay.
The process of
writing an algebraic expression as a product of its factors is called factorization
of the algebraic expression.
Methods of Factorization
We have the following methods of
factorizing algebraic expressions.
1. Finding the common factor of an algebraic
expression when each of its terms contains a common monomial factor.
2. Finding factors by arranging the
terms of an algebraic expression into groups which have a factor common
to them.
3. Finding factors of the difference
of two squares.
Factorization by Taking out Common Factors
When the terms of a given algebraic
expression have common factors, we factorize the expression by the
following procedure.
1. Find the greatest or highest common
factor (HCF) of all the terms of the expression.
2. Divide the terms by the HCF,
enclose the quotients within a bracket and keep the common factor outside the
bracket.
Example 1: Factorize the following:
a. 15a + 20b b. 24ax – 40ay + 8a
Solution: a. 15a + 20b = 5(3a) + 5(4b)
= 5(3a + 4b)
b. 24ax – 40ay + 8a = (8a)(3x) –
(8a)(5y) + (8a)(1)
=
8a(3x – 5y + 1)
Factorization by Grouping the Terms
In this method, an algebraic expression
with common factors can be found out by grouping its terms in a suitable
arrangement. Follow the steps given below to factorize an algebraic expression
by grouping the terms.
1. Group the terms of the given
algebraic expression such that each group has a common factor.
2. Take out the common factors.
Example 2: Factorize the following:
a. 12ax – 3ay + 8bx – 2by b. 49a + 42c –
7ay – 6cy
Solution: a. 12ax – 3ay + 8bx – 2by = (12ax – 3ay) +
(8bx – 2by)
= 3a(4x – y) + 2b(4x – y)
= (3a + 2b)(4x – y)
b. 49a + 42c – 7ay – 6cy = (49a – 7ay) + (42c – 6cy)
= 7a(7 – y) + 6c(7 – y)
= (7 – y) (7a + 6c)
b. 49a + 42c – 7ay – 6cy = (49a – 7ay) + (42c – 6cy)
= 7a(7 – y) + 6c(7 – y)
= (7 – y) (7a + 6c)
Factorization by Finding the Difference between Two Squares
When an algebraic expression is
expressed as the difference between two squares, we can factorize the algebraic
expression by the relation a2 – b2 = (a + b)(a – b).
Thus, the factors of a2 – b2
are (a + b) and (a – b).
Example 3: Factorize the following:
a. 9x2 – 16y2 b. 64x2
– 36
Solution: a. 9x2 – 16y2 = (3x)2
– (4y)2
= (3x
+ 4y) (3x – 4y)
b. 64x2 – 36 = (8x)2
– (6)2
= (8x + 6) (8x – 6)
Factorization of Quadratic Trinomials
An algebraic expression of the form ax2
+ bx + c is called a quadratic expression.
Since the algebraic expression ax2
+ bx + c has three terms, it is also known as a quadratic trinomial.
Case 1: When trinomial is of the form x2
+ bx + c, means a = 1, then we find two integers l and m such that (l + m) = b
and lm = c.
Therefore, x2 + bx + c = (x
+ l) (x + m).
Case 2: When trinomial is of the form ax2
+ bx + c and a ˃ 1, then we find two integers l and m such that (l + m) = b and
lm = ac.
Example 4: Factorize the following:
a. x2 + 7x + 12 b. 14x2
– 23x + 8
Solution: a. x2 + 7x + 12 = x2 +
(4 + 3)x + 4 × 3
= x2
+ 4x + 3x + 12
=
x(x + 4) + 3(x + 4)
= (x
+ 4)(x + 3)
b. 14x2 – 23x + 8 = 14x2
– (7 + 16)x + 8
= 14x2 – 7x – 16x + 8
= 7x (2x – 1) –
8 (2x – 1)
= (2x – 1) (7x –
8)