Subject: Compulsory Maths

Factorization is the process of expressing the given algebraic expression as the product of two or more algebraic expressions or terms.

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

- (a+b)
^{2 }= a^{2}+2ab+b^{2}or (a+b)^{2}+4ab - (a-b)
^{2 }= a^{2}-2ab+b^{2}or, (a+b)^{2}-4ab - a
^{2}-b^{2 }= (a+b)(a-b) - a
^{2}+b^{2}= (a+b)^{2}-2ab or, (a-b)^{2}+2ab

- 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.

Factorize:

16x^{4}-4x^{2}

Soln:16x^{4}-4x^{2}

= 4x^{2}(4x^{2}-1)

= 4x^{2}{(2x^{2})(1)^{2}}

=4x^{2}(2x+1)(2x-1) Ans.

Factorize:

3x^{3}-48x

Soln:3x^{3}-48x

= 3x(x^{2}-16)

= 3x(x^{2}-4^{2})

=3x(x+4)(x-4) Ans.

Factorize:

a^{4}-b^{4}

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.

Factorize:

(y+1)^{2}-4

Soln:(y+1)^{2}-4

=(y+1)^{2}-(2)^{2}

=(y+1+2)(y+1-2)

= (y+3)(y-1)Ans.

Factorize:

36-(x+5)^{2}

Soln:36-(x+5)^{2}

= 6^{2}-(x+5)^{2}

=(6+x+5)(6-x-5)

=(11+x)(1-x) Ans.

Factorize:

a^{2}+2ab+b^{2}-c^{2}

Soln:a^{2}+2ab+b^{2}-c^{2}

= (a+b)^{2}-c^{2}

=(a+b+c)(a+b-c) Ans.

Factorize:

m^{2}+n^{2}-2mn-p^{2}

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.

Factorize:

ab^{2}-b(a-c)-c

Soln:ab^{2}-b(a-c)-c

= ab^{2}-ab+bc-c

= ab(b-1) +c (b-1)

= (b-1)(ab+c) Ans.

Factorize:

x^{2}-2(x+y)-y^{2}

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.

Simplify by using formula:

(32)^{2}-(28)^{2}

Soln:(32)^{2}-(28)^{2}

= (32+28)(32-28)

= 60 ×4

= 240 Ans.

Simplify by using formula:

(51)^{2} -(49)^{2}

Soln:(51)^{2} -(49)^{2}

= (51+49)(51-49)

= 100× 2

= 200 Ans.

Simplify by using formula:

(101)^{2}-(99)^{2}

Soln: (101)^{2}-(99)^{2}

= (101+99)(101-99)

= 200 × 2

=400 Ans.

Simplify by using formula:

(7.9)^{2}-(2.1)^{2}

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

Resolve into factors:

4x^{2}+9x+5

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.

Resolve into factors:

4x^{2}-12x+5

Soln:4x^{2}-12x+5

= 4x^{2}-10x -2x+5

= 2x(2x-5)-1(2x-5)

=(2x-1)(2x-5) Ans.

Factorize the following expressions:

p^{4}+4

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.

Factorize the following expressions:

a^{4}+4

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.

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