 ## Factorization

Subject: Compulsory Maths

#### Overview

Factorization is the process of finding the factors. Factoring is the decomposition of an object, into a product of other objects, or factors, which when multiplied together give the original. This note contains information about factorization.

##### Factorization

When two or more algebraic expressions are multiplied, the result is called product and each expression is called the factor of the product.

The process of finding out factors of an algebraic expression is known as factorisation.

For example:

If we factorise (bc + cd), you get c ( b + d ).

### Factorizing the difference of two squares

Let's multiply ( a + b ) and ( a - b )

( a + b ) ( a - b )

= a² - ab + ab -b²

= a² - b² ( This expression is called a difference of two squares )

Therefore, the factors of a² - b² are ( a + b ) and ( a - b)

Examples:

1. x- 49
Solution:
x- 49, this expression is the difference of two squares.
= x2- 72, which is in the form of a2- b2
= (x+7) (x-7)

2. 4y- 36y6
Solution:
In 4y- 36y6, there is a common factor of 4y2 that can be factored out first in this problem, to make the problem easier.
= 4y- 36y6
= 4y2(1 - 9y4)
= 4y2{(1)2- (3y2)2}
= 4y2(1+3y2)(1-3y2)

### Factoring perfect square trinomials

Let's multiply (a+b) and (a+b)  (a+b) (a+b)

= a2+ ab + ab + b2

= a2 + 2ab + b2

Thus, a2 + 2ab + b2 = (a + b)2 and (a + b)2 is the factorisation form of a+ 2ab + b2

Similarly, a- 2ab + b2 = (a -b)2 and (a - b)2 is the factorisation form of a- 2ab +b= (a -b)and (a - b)2 is the factorisation form of a2 - 2ab + b2

### Geometrical meaning

If we consider (a+ b) as one of the side of the square then the product of the expression will form two squares namely a2 and b2 and two congruent rectangles with each having an area of ab.

 a2 ab ab b2

Area of the entire square = (a + b)2

Area of two squares and two rectangles

= a2 + ab +ab + b2

= a2 + 2ab +b2

Thus, a+ 2ab + b2 = (a+b)2

##### Things to remember
• Factorization is the process of finding the factors.
• Factoring is the decomposition of an object, into a product of other objects, or factors, which when multiplied together give the original.
• It includes every relationship which established among the people.
• There can be more than one community in a society. Community smaller than society.
• It is a network of social relationships which cannot see or touched.
• common interests and common objectives are not necessary for society.

Solution:

y2-9, this expression is the difference of two squares.

=y2-32, which is in the form of a2-b2

=(y+3)(y-3)

Solution:

In 4y2-366, there is a common factor of 4y2 that can be factored out first in this problem, to make the problem easier.

4y2- 36y6

= 4y2(1 - 9y4)

= 4y2{(1)2- (3y2)2}

= 4y2(1 + 3y2)(1 - 3y2)

Solution:

Given expression =6x+3

= 2.3.x + 3

= 3(2x+1) [3 is common in both]

Solution:

Given expression =x2+4x

=x. x+4. x

=x(x+4) [x is common in both]

Solution:

Given expression =12a+3 b

=4.3. a + 3.b

=3(4a+b) [ 3 is common in both]

Solution:

Here,

Given expression =x+x3

=x+x . x.x

=x(1+x2) [x is common in both]

Solution:

Here,

Given expression =12x2+xy+xz

=2.2.3.x.x+x.y+x.z

=x(12x + y + z) [ x is common in all]

Solution:

Here,

Given expression =14xy+7y

=2.7.x.y+7.y

=7y(2x+1) [ 7y is common in both]

Solution:

9a2+24ab+16b2

Since 24 =2×3×4 and 9=32,16=42,

9a2+24ab+16b2=(3a)2+2.3a.4b+(4b)2

=(3a)2+2.3a.4b+(4b)2

=(3a+4b)2

Solution:

25a2-80a+64

Since 80=2×5×8, 25=52 and 64=82,

25a2-80a+64=(5a)2-2.5a.8+(8)2

= (5a-8)2

Solution:

Comparing x2+5x+6 with x2+(a+b) x+ab we get,

a+b =5 and ab=6

So, 3+2=5 and 3×2=6

Hence, x2+5x+6=x2+(3+2)x+6

=(x+3)(x+2)

 ab=6 a+b=5 1×6=6 1+6=7 2×3=6 2+3=5

Solution:

For a2+7a-18, two numbers whose difference is 7 and product-18 are 9 and -2

Hence, a2+7a-18=a2+(9-2)a-18

= (a+9)(a-2)

Solution:

For b2-7b-18, two numbers whose sum is -7 and product -18 are -9 and 2

Hence, b2-7b-18

=b2(-9+2)b-18

= (b-9)(b+2)

Solution:

64x3+125=(4x)3+53

=(4x+5)[(4x)2-4x.5+52]

=(4x+5)(16x2-20x+25)

Solution:

3a3-81=3(a3-27)

=3(a3-33)

=3(a-3)(a2+3a+9)

Solution:

Comparingx2-7x=12 with x2-(a+b) x+ab we get,

a+b=7 and ab =12

So, 4+3=7 and 4×3=12

Hence, x2-7x+12=x2-(4+3) x+12

=x2+(-4-3)x + 12

= (x-4)(x-3)

 ab=12 a+b=7 12×1=12 12+1=13 6×2=12 6+2=8 4×3=12 4+3=7

Solution:

Here givenx2+3x+xy+3y

=x(x+3)+y(x+3)

=(x+3)(x+y) Ans.

Solution:

Here,

x2- 4

=(x)2-(2)2

=(x-2)(x+2) [$\therefore$a2-b2=(a+b)(a-b)]

Solution:

Here,

Given = 9x2-y2

=(3x)2-(y)2

=(3x+y)(3x-y)

Solution:

Given = 121-25y2

= (11)2-(5y)2

= (11+5y)(11-5y)

Solution:

The given expression is x2 - 7x + 12

Find two numbers whose sum = -7 and product = 12

Clearly, such numbers are (-4) and (-3).

Now, x2 - 7x + 12

= x2 - 4x - 3x + 12

= x(x - 4) -3 (x - 4)

= (x - 4)(x - 3)

Solution:

The given expression is 15x2 - 26x + 8.

Find two numbers whose sum = -26 and product = (15 × 8) = 120.

Clearly, such numbers are -20 and -6.

Here, 15x2 - 26x + 8

= 15x2 - 20x - 6x + 8

= 5x(3x - 4) - 2(3x - 4)

= (3x - 4)(5x - 2)