**Factorization**

In arithmetic, some positive integers
can be written as a product of their prime factors.

For example, 600 = 2

^{3}× 3 × 5^{2}
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 a

^{2}– b^{2}= (a + b)(a – b).
Thus, the factors of a

^{2}– b^{2}are (a + b) and (a – b).**Example 3:**Factorize the following:

a. 9x

^{2}– 16y^{2}b. 64x^{2}– 36**Solution:**a. 9x

^{2}– 16y

^{2}= (3x)

^{2}– (4y)

^{2}

= (3x
+ 4y) (3x – 4y)

b. 64x

^{2}– 36 = (8x)^{2}– (6)^{2}
= (8x + 6) (8x – 6)

**Factorization of
Quadratic Trinomials**

An algebraic expression of the form ax

^{2}+ bx + c is called a quadratic expression.
Since the algebraic expression ax

^{2}+ bx + c has three terms, it is also known as a quadratic trinomial.**Case 1**: When trinomial is of the form x

^{2}+ bx + c, means a = 1, then we find two integers l and m such that (l + m) = b and lm = c.

Therefore, x

^{2}+ bx + c = (x + l) (x + m).**Case 2**: When trinomial is of the form ax

^{2}+ 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. x

^{2}+ 7x + 12 b. 14x^{2}– 23x + 8**Solution:**a. x

^{2}+ 7x + 12 = x

^{2}+ (4 + 3)x + 4 × 3

= x

^{2}+ 4x + 3x + 12
=
x(x + 4) + 3(x + 4)

= (x
+ 4)(x + 3)

b. 14x

^{2}– 23x + 8 = 14x^{2}– (7 + 16)x + 8
= 14x

^{2}– 7x – 16x + 8
= 7x (2x – 1) –
8 (2x – 1)

= (2x – 1) (7x –
8)