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Note on Algebraic Expresssion

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Factorization

Factorization is the process of expressing the given algebraic expression as the product of two or more algebraic expression or terms. For example, 2x+5x2  = x(2 + 5x) can be expressed as the product of x and 2 +5x.i.e 2x+52 = 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+5x2.

Factorization of the expression of the form a2-b2

(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 a2-b2 diagonally as shown in the figure.

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(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)

= a2-b2

∴ Area of rectangle = (a2-b2) sq.unit

∴ a2-b2 = (a+b)(a-b)

Some useful formula

  • (a+b)= a2+2ab+b2 or (a+b)2+4ab
  • (a-b)2 = a2-2ab+b2 or, (a+b)2-4ab
  • a2-b= (a+b)(a-b)
  • a2+b2 = (a+b)2-2ab or, (a-b)2+2ab
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Very Short Questions

Soln:16x4-4x2

= 4x2(4x2-1)

= 4x2{(2x2)(1)2}

=4x2(2x+1)(2x-1) Ans.

Soln:3x3-48x

= 3x(x2-16)

= 3x(x2-42)

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

Soln: a4-b4

=(a2)2-(a2)2-(b2)2

=(a2+b2)(a2-b2)

=(a2+b2)(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

= 62-(x+5)2

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

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

Soln:a2+2ab+b2-c2

= (a+b)2-c2

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

Soln: m2+n2-2mn-p2

=m2-2mn+n2-p2

=(m -n)2-p2

= (m-n+p)(m-n-p) Ans.

Soln:ab2-b(a-c)-c

= ab2-ab+bc-c

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

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

Soln:x2-2(x+y)-y2

= x2-y2-2(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: 4x2+9x+5

= 4x2+5x +4x+5

= 4x2+5x+4x+5

= x(4x+5)+1(4x+5)

=(x+1)(4x+5) Ans.

Soln:4x2-12x+5

= 4x2-10x -2x+5

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

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

Soln:p4+4

=(p2)2+22

=(p2)2+2.p2.2+22-4p2

=(p2+2)2-4p2=(p2+2)2-(2p)2

=(p2+2+2p)(p2+2-2p)

=(p2+2p+2)(p2-2p+2) Ans.

Soln:a4+4

=(a2)2+22

=(a2)2+2.a2.2+22-4a52

=(a2+2)2-4a2=(a2+2)2-(2a)2

=(a2+2+2a)(a2+2-2a)

=(a2+2a+2)(a2-2a+2) Ans.

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  • x2-y2-ax+ay

    (x-y)(x-y-a)


    (x-y)(x+y-a)


    (x-y)(x+y+a)


    (x+y)(x+y-a)


  • a2+6ab-c2-6bc

    (a-b) (a+6b+c)


    (a+c) (a+6b+c)


    (a-c) (a+6b+c)


    (a+6b) (a+6b+c)


  • a2-2a-2b-b2

    (a-b) (a-b-2)


    (a+b) (a-b+2)


    (a+b) (a-b-2)


    (a+b) (a-a-2)


  • a2-2a-2b-b2

    (a+b) (a-b+2)


    (a+b) (a-a-2)


    (a+b) (a-b-2)


    (a-b) (a-b-2)


  • b2-ab+a2-ab

    (a+b) (a+b)


    (a-b) (a+b)


    (a+b) (a-b)


    (a-b) (a-b)


  • x3-x2-x+1

    (x-1) (x-1) (x+1)


    (x-1) (x+1) (x+1)


    (x-2) (x-1) (x+1)


    (x-1) (x-1) (x+3)


  • 8-36x-2x2+9x3

    (x-2) (x+2) (9x-2)


    (x-2) (x+2) (9x+2)


    (x+2) (x+2) (9x-2)


    (x-2) (x-2) (9x-2)


  • a4+a2+1

    (a2+a-1) (a2-a+1)


    (a3+a+1) (a2-a+1)


    (a2+a+2) (a2-a+1)


    (a2+a+1) (a2-a+1)


  • m4+m2+1

    (m2+m+1) (m2-m-1)


    (m2+m+1) (m2-m+2)


    (m2+m+1) (m2-m+1)


    (m2+m-1) (m2-m+1)


  • 4x2-y2+2y-1

    (2x+y+1) (2x-y+1)


    (2x+y-1) (2x-y-1)


    (2x+y-1) (3x-y+1)


    (2x+y-1) (2x-y+1)


  • 9a2-4b2+6a+1

    (3a+2b+1) (4a-2b+1)


    (3a+2b+1) (3a+2b+1)


    (3a+2b+1) (3a-2b-1)


    (3a+2b+1) (3a-2b+1)


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DISCUSSIONS ABOUT THIS NOTE

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Krischal Dhungel

(a-b)x^2 - (a b)x 2b = 0


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Krischal Dhungel

if x^1/m=y^1/n = z^1/p then prove that : m n p=0


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