## Note on Factorization

• Note
• Things to remember
• Exercise
• Quiz

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

• 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.
.

### Very Short Questions

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:

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)

0%

9 and 2
1 and 1
3 and 5
3 and 1

21 and 1
7 and 3
20 and 4
9 and 5

5x
10x
9x
11y

2 y
10 y
10 y
18 y

12 xy
5 xy
2 xy
10 xy

2 x
24 x
20 x
25 x

50 x
80 x
20 x
30 x

35 y
25 y
60 y
50 y

2600
2100
2800
2550

980
900
250
820

100000
150000
110000
120000

56
84
22
63
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