Factorization, Methods of Factorization

# Factorization, Methods of Factorization

## 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)

## 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)

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)