Factorization is the process of expressing the given algebraic expression as the product of two or more algebraic expression or terms. For example, 2x+5x^{2 }= x(2 + 5x) can be expressed as the product of x and 2 +5x.i.e 2x+5^{2} = x(2 + 5x).When an algebraic expression is the product of two or more expressions, each of these expressions is called a factor of a given expression.In the above examples, x and 2x+5x are the factors of 2x+5x^{2}.
Factorization of the expression of the form a^{2}-b^{2}
(i) Take a square piece of paper with one side 'a' unit whose area becomes as sq.units.
(ii) In one of the corners of the square, cut off a small square of side 'b' units. The area of this small squares is so units.
(iii) Let's cut a^{2}-b^{2} diagonally as shown in the figure.
(iv) Arrange two parts to make a rectangle.
Length of the rectangle = (a+b) units
Breagth of the rectangle = (a-b)
Now, area of the rectangle = length × breadth
= (a+b)(a-b)
= a^{2}-b^{2}
∴ Area of rectangle = (a^{2}-b^{2}) sq.unit
∴ a^{2}-b^{2 }= (a+b)(a-b)
Some useful formula
Soln:16x^{4}-4x^{2}
= 4x^{2}(4x^{2}-1)
= 4x^{2}{(2x^{2})(1)^{2}}
=4x^{2}(2x+1)(2x-1) Ans.
Soln:3x^{3}-48x
= 3x(x^{2}-16)
= 3x(x^{2}-4^{2})
=3x(x+4)(x-4) Ans.
Soln: a^{4}-b^{4}
=(a^{2})^{2}-(a^{2})^{2}-(b^{2})^{2}
=(a^{2}+b^{2})(a^{2}-b^{2})
=(a^{2}+b^{2})(a+b)(a-b) Ans.
Soln:(y+1)^{2}-4
=(y+1)^{2}-(2)^{2}
=(y+1+2)(y+1-2)
= (y+3)(y-1)Ans.
Soln:36-(x+5)^{2}
= 6^{2}-(x+5)^{2}
=(6+x+5)(6-x-5)
=(11+x)(1-x) Ans.
Soln:a^{2}+2ab+b^{2}-c^{2}
= (a+b)^{2}-c^{2}
=(a+b+c)(a+b-c) Ans.
Soln: m^{2}+n^{2}-2mn-p^{2}
=m^{2}-2mn+n^{2}-p^{2}
=(m -n)^{2}-p^{2}
= (m-n+p)(m-n-p) Ans.
Soln:ab^{2}-b(a-c)-c
= ab^{2}-ab+bc-c
= ab(b-1) +c (b-1)
= (b-1)(ab+c) Ans.
Soln:x^{2}-2(x+y)-y^{2}
= x^{2}-y^{2}-2(x+y)
= (x+y)(x-y)-2(x+y)
=(x+y)(x-y-2) Ans.
Soln:(32)^{2}-(28)^{2}
= (32+28)(32-28)
= 60 ×4
= 240 Ans.
Soln:(51)^{2} -(49)^{2}
= (51+49)(51-49)
= 100× 2
= 200 Ans.
Soln: (101)^{2}-(99)^{2}
= (101+99)(101-99)
= 200 × 2
=400 Ans.
Soln:(7.9)^{2}-(2.1)^{2}
=(7.9)^{2}-(2.1)^{2}
=(7.9 +2.1)(7.9-2.1)
= 10.0 × 5.8
= 58 Ans
Soln: 4x^{2}+9x+5
= 4x^{2}+5x +4x+5
= 4x^{2}+5x+4x+5
= x(4x+5)+1(4x+5)
=(x+1)(4x+5) Ans.
Soln:4x^{2}-12x+5
= 4x^{2}-10x -2x+5
= 2x(2x-5)-1(2x-5)
=(2x-1)(2x-5) Ans.
Soln:p^{4}+4
=(p^{2})^{2}+2^{2}
=(p^{2})^{2}+2.p^{2}.2+2^{2}-4p^{2}
=(p^{2}+2)^{2}-4p^{2}=(p^{2}+2)^{2}-(2p)^{2}
=(p^{2}+2+2p)(p^{2}+2-2p)
=(p^{2}+2p+2)(p^{2}-2p+2) Ans.
Soln:a^{4}+4
=(a^{2})^{2}+2^{2}
=(a^{2})^{2}+2.a^{2}.2+2^{2}-4a5^{2}
=(a^{2}+2)^{2}-4a^{2}=(a^{2}+2)^{2}-(2a)^{2}
=(a^{2}+2+2a)(a^{2}+2-2a)
=(a^{2}+2a+2)(a^{2}-2a+2) Ans.
x^{2}-y^{2}-ax+ay
(x-y)(x+y-a)
(x-y)(x+y+a)
(x-y)(x-y-a)
(x+y)(x+y-a)
a^{2}+6ab-c^{2}-6bc
(a-c) (a+6b+c)
(a-b) (a+6b+c)
(a+c) (a+6b+c)
(a+6b) (a+6b+c)
a^{2}-2a-2b-b^{2}
(a+b) (a-a-2)
(a+b) (a-b+2)
(a-b) (a-b-2)
(a+b) (a-b-2)
a^{2}-2a-2b-b^{2}
(a+b) (a-b-2)
(a-b) (a-b-2)
(a+b) (a-a-2)
(a+b) (a-b+2)
b^{2}-ab+a^{2}-ab
(a+b) (a-b)
(a-b) (a+b)
(a+b) (a+b)
(a-b) (a-b)
x^{3}-x^{2}-x+1
(x-1) (x+1) (x+1)
(x-2) (x-1) (x+1)
(x-1) (x-1) (x+1)
(x-1) (x-1) (x+3)
8-36x-2x^{2}+9x^{3}
(x-2) (x-2) (9x-2)
(x-2) (x+2) (9x+2)
(x-2) (x+2) (9x-2)
(x+2) (x+2) (9x-2)
a^{4}+a^{2}+1
(a^{3}+a+1) (a^{2}-a+1)
(a^{2}+a+1) (a^{2}-a+1)
(a^{2}+a+2) (a^{2}-a+1)
(a^{2}+a-1) (a^{2}-a+1)
m^{4}+m^{2}+1
(m^{2}+m-1) (m^{2}-m+1)
(m^{2}+m+1) (m^{2}-m-1)
(m^{2}+m+1) (m^{2}-m+2)
(m^{2}+m+1) (m^{2}-m+1)
4x^{2}-y^{2}+2y-1
(2x+y-1) (2x-y-1)
(2x+y-1) (3x-y+1)
(2x+y-1) (2x-y+1)
(2x+y+1) (2x-y+1)
9a^{2}-4b^{2}+6a+1
(3a+2b+1) (3a-2b-1)
(3a+2b+1) (3a-2b+1)
(3a+2b+1) (4a-2b+1)
(3a+2b+1) (3a+2b+1)
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Jul 01, 2017
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Krischal Dhungel
(a-b)x^2 - (a b)x 2b = 0
Mar 10, 2017
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Krischal Dhungel
if x^1/m=y^1/n = z^1/p then prove that : m n p=0
Mar 10, 2017
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