Note on Factorization

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  • Things to remember
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  • Quiz

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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

Factoring perfect square trinomials
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.
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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%
  • Find the value of a and b for:

    ab=18, a + b =11

    9 and 2
    1 and 1
    3 and 5
    3 and 1
  • Find the value of a and b for:

     ab=21, a + b=22

    21 and 1
    7 and 3
    20 and 4
    9 and 5
  • What should be filled in the given blank in order to make the expression a perfect square?

    x2+______+25

    5x
    10x
    9x
    11y
  • What should be filled in the given blank in order to make the expression a perfect square?

     y2- ______+81

    2 y
    10 y
    10 y
    18 y
  • What should be filled in the gap below in order to make the expression a perfect square?

     4x2 +______+ 9y2

    12 xy
    5 xy
    2 xy
    10 xy
  • What should be filled in the gap below in order to make the expression a perfect square?

     9x2 + ______ + 16

    2 x
    24 x
    20 x
    25 x 
  • What should be filled in the given blank in order to make the expression a perfect square?

     25x2- ______ + 64

    50 x
    80 x
    20 x
    30 x
  • What should be filled in the given blank in order to make the expression a perfect square?

     36 - ______ + 25y2

    35 y
    25 y
    60 y
    50 y 
  • What is the correct value of (64)2 – (36)2?

    2600
    2100
    2800
    2550
  • Which one of the following is the correct value of (42)2 – (28)2?

    980
    900
    250
    820
  • Find the value of:

    (10003)2 – (9997)2

    100000
    150000
    110000
    120000
  • Which one of them is the correct value of (9.2)2 – (0.8)2?

    56
    84
    22
    63
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DISCUSSIONS ABOUT THIS NOTE

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Forum Time Replies Report
8-8x 6


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factorization

a-b


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shriyash amatya

5ab


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raju gupta

a 2ab


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