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 ).
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:
Let's multiply (a+b) and (a+b)
(a+b) (a+b)
= a^{2}+ ab + ab + b^{2}
= a^{2} + 2ab + b^{2}
Thus, a^{2} + 2ab + b^{2} = (a + b)^{2} and (a + b)^{2} is the factorisation form of a^{2 }+ 2ab + b^{2}
Similarly, a^{2 }- 2ab + b^{2} = (a -b)^{2} and (a - b)^{2} is the factorisation form of a^{2 }- 2ab +b^{2 }= (a -b)^{2 }and (a - b)^{2} is the factorisation form of a^{2} - 2ab + b^{2}
If we consider (a+ b) as one of the side of the square then the product of the expression will form two squares namely a^{2} and b^{2} and two congruent rectangles with each having an area of ab.
a^{2} | ab |
ab | b^{2} |
Area of the entire square = (a + b)^{2}
Area of two squares and two rectangles
= a^{2} + ab +ab + b^{2}
= a^{2} + 2ab +b^{2}
Thus, a^{2 }+ 2ab + b^{2} = (a+b)^{2}
Solution:
Given expression =6x+3
= 2.3.x + 3
= 3(2x+1) [3 is common in both]
Solution:
Given expression =x^{2}+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+x^{3}
=x+x . x.x
=x(1+x^{2}) [x is common in both]
Solution:
Here,
Given expression =12x^{2}+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,
x^{2}- 4
=(x)^{2}-(2)^{2}
=(x-2)(x+2) [\(\therefore\)a^{2}-b^{2}=(a+b)(a-b)]
Solution:
Here,
Given = 9x^{2}-y^{2}
=(3x)^{2}-(y)^{2}
=(3x+y)(3x-y)
Solution:
Given = 121-25y^{2}
= (11)^{2}-(5y)^{2}
= (11+5y)(11-5y)
Solution:
The given expression is x^{2} - 7x + 12
Find two numbers whose sum = -7 and product = 12
Clearly, such numbers are (-4) and (-3).
Now, x^{2} - 7x + 12
= x^{2} - 4x - 3x + 12
= x(x - 4) -3 (x - 4)
= (x - 4)(x - 3)
Find the value of a and b for:
ab=18, a + b =11
Find the value of a and b for:
ab=21, a + b=22
What should be filled in the given blank in order to make the expression a perfect square?
x^{2}+______+25
What should be filled in the given blank in order to make the expression a perfect square?
y^{2}- ______+81
What should be filled in the gap below in order to make the expression a perfect square?
4x^{2} +______+ 9y^{2}
What should be filled in the gap below in order to make the expression a perfect square?
9x^{2} + ______ + 16
What should be filled in the given blank in order to make the expression a perfect square?
25x^{2}- ______ + 64
What should be filled in the given blank in order to make the expression a perfect square?
36 - ______ + 25y^{2}
What is the correct value of (64)^{2} – (36)^{2}?
Which one of the following is the correct value of (42)^{2} – (28)^{2}?
Find the value of:
(10003)^{2} – (9997)^{2}
Which one of them is the correct value of (9.2)^{2} – (0.8)^{2}?
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