Useful formulae

(i)     (a + b)2 = a2 + 2ab + b2

(ii)    (a – b)2 = a2 – 2ab + b2

(iii)   a2 – b2 = (a + b) (a – b)                                   

(v)    a3 + b3 = (a + b) (a2 – ab + b2)

(vi) a3 + b3 = (a + b)3 – 3a(a + b)                           

(vii)  a3 – b3 = (a – b) (a2 + ab + b2)

(viii) a3 – b3 = (a – b)3 + 3ab (a – b)

(ix)   (a + b)3 = a3 + 3a2b + 3ab2 + b3

 (ix)  (a + b)3 = a3 + b3 + 3ab (a + b)                      

(x) (a – b)3 = a3 – 3a2b + 3ab2 – b3

 (xi) (a – b)3 =  a3 – b3 – 3ab(a – b)

(xii)  (a + b + c)2 = a2 + b2 + c2 + 2ab + 2bc + 2ca              

(xiii) (a – b – c)2 = a2 + b2 + c2 – 2ab + 2bc + 2ca              

(xiv) a4 + a2b2 + b4 = (a2 + ab + b2) (a2 – ab + b2)

 Factorize

Factorization

The process of finding factors is called factorization.

The factors of 12 are 1, 2, 3, 4, 6 and 12

The prime factors of 12 are 2, 2, 3

Factorization of an algebraic expression

The process of finding factors of algebraic expression is called factorization of algebraic expression.

3x2 = 3 ´ x ´ x

Things to know while doing factorization of an algebraic expression

ü    Check whether there is any common factor on each term. If same factor is there in each term, then take that as common factor.

ü    Rearrange the term, (if necessary) and take common.

ü  Use formula if necessary

ü Split the term or Break the term

Q.1.        Factorize: x2 – 2x

Soln:   Here, x2 – 2x = x(x – 2)   Ans.

Q.2.        Factorize: 3a2 – 12ab

Soln:   Here, 3a2 – 12ab = 3a(a – 4b)  Ans.

Q.3.        Factorize: ab + bc + ax + cx

Soln:   Here, ab + bc + ax + cx = b(a + c) + x(a + c)

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

Q.4.        Write the factors of x2 – 25

Soln:   Here, x2 – 25 = x2 – 52

                       = (x – 5)(x + 5)   Ans.

Q.5.        Factorize: x(x – y)2 + 2xy(x – y)

Soln:   Here, x(x – y)2 + 2xy(x – y) = x(x – y) {(x – y) + 2y}

                                                     = x(x – y) (x – y + 2y)

                                             = x(x – y)(x + y)   Ans.

Q.6.        Factorize): 4a4 + x4

Soln:   Here, 4a4 + x4 = (2a2)2 + (x2)2

                     = (2a2 + x2)2 – 2.2a2.x2

                    = (2a2 + x2)2 – 4a2x2

                      = (2a2 + x2)2 – (2ax)2

                      = {(2a2 + x2) – 2ax}{(2a2 + x2) + 2ax}

                       = (2a2 – 2ax + x2) (2a2 + 2ax + x2)   Ans.

Q.7.        Factorize: p4 + 64q4

Soln:   Here, p4 + 64q4 = (p2)2 + (8q2)2

                           = (p2 + 8q2)2 – 2p2.8p2

                           = (p2 + 8q2)2 – 16p2q2

                           = (p2 + 8q2)2 – (4pq)2

                           = (p2 + 8p2 – 4pq) (p2 + 8q2 + 4pq)

                            = (p2 – 4pq + 8q2) (p2 + 4pq + 8q2)   Ans.

Q.9.               Factorize: x4 + 5x2 + 9

Soln:   Here, x4 + 5x2 + 9 = x4 + 9 + 5x2

                             = (x2)2 + 32 + 5x2

                             = (x2 + 3)2 – 2x2.3 + 5x2

                            = (x2 + 3)2 – x2

                             = {(x2 + 3) – x} {(x2 + 3) + x}

                             = (x2 – x + 3) (x2 + x + 3)   Ans.


Q.10.        Factorize: p2 + 2p – 8 – q2 – 6q

Soln:   Here, p2 + 2p – 8 – q2 – 6q

        = p2 + 2p + 1 – 9 – q2 – 6q

        = p2 + 2.p.1 + 12 – (q2 + 6q + 9)

        = (p + 1)2 – (q2 + 2.q.3 + 32)

        = (p + 1)2 – (q + 3)2

        = {(p + 1) – (q + 3)}{(p + 1) + (q + 3)}

        = (p – q – 2) (p + q + 4)   Ans.

Q.11.        Factorize: 4m2 – 12m – 16 – 9x2 + 30n

Soln:   Here, 4m2 – 12m – 16 – 9x2 + 30n

                      = 4m2 –12m + 9 – 25 – 9n2 + 30n

                      = (2m)2 – 2.2m.3 + 32 – (9x2 – 30n + 25)

                      = (2m – 3)2 – {(3n)2 – 2.3n.5 + 52)

                      = (2m – 3)2 – (3n – 5)2

                      = {(2m – 3) – (3n – 5)}{(2m – 3) + (3n – 5)}

                      = (2m – 3n + 2) (2m + 3n – 8)  Ans.

Q.12.        Factorize: a2 – 6a + 8 – b2 + 2b

Soln:   Here, a2 – 6a + 8 – b2 + 2b

                      = a2 – 6a + 9 – 1 – b2 + 2b

                      = a2 – 2.a.3 + (3)2 – (b2 – 2b + 1)

                      = (a – 3)2 – (b – 1)2

                      = {(a – 3) + (b – 1)}{(a – 3) – (b – 1)}

                      = (a + b – 4) (a – b – 2)   Ans.

Q. 13.         y4 + y2 + 1                   [Use the formula a2 + b2 = (a + b)2 - 2ab      or, (a - b)2 + 2ab]

Solution

Here,

             y4 + y2 + 1 = y4 + 1 + y2

                                = (y2)2 + (1)2 + y2

                                = (y2 + 1)2 - 2.y2.1 + y2

                                = (y2 + 1)2 - 2y2 + y2

                                = (y2 + 1)2 - y2

                                = {(y2 + 1) + y}{(y2 + 1) - y}

                                = (y2 + y + 1)(y2 - y + 1) Ans

Q.13.         x4 + x2y2 + y4

Solution:

Here, x4 + x2y2 + y4 = x4 + y4 + x2y2

                                      = (x2)2 + (y2)2 + x2y2

                                      = (x2 + y2)2 – 2.x2.y2 + x2y2

                                      = (x2 + y2)2 – 2x2y2 + x2y2

                                      = (x2 + y2)2 – x2y2

                                      = (x2 + y2)2 – (xy)2

                                      = {(x2 + y2) + xy} {(x2 + y2) - xy}

                                      = (x2 + xy + y2)(x2 - xy + y2) Ans.

 

 

 

Q.14.         x8 + x4 + 1

Solution

Here,

             x8 + x4 + 1 = x8 +1 + x4

                                = (x4)2 + (1)2 + x4

                                = (x4 + 1)2 - 2.x4.1 + x4

                                = (x4 + 1)2 - 2x4 + x4

                                = (x4 + 1)2 - x4

                                = (x4 + 1)2 - (x2)2

                                = {(x4 + 1) + x2}{(x4 + 1) - x2}

                                = (x4 + 1 + x2)(x4 -  x2 + 1)

                                = {(x2)2 + (1)2 + x2}(x4 -  x2 + 1)

                                = {(x2 + 1)2 - 2x2.1 + x2}(x4 -  x2 + 1)

                                = {(x2 +1)2 - x2}(x4 -  x2 + 1)

                                = {(x2 +1) + x}{(x2 +1) - x} (x4 -  x2 + 1)

                                = (x2 + x + 1)(x2 - x + 1} (x4 -  x2 + 1)

Q. 15.         x4 + 4y4

Solution a2 + b2 = (a + b)2 - 2ab or (a - b)2 + 2ab

Here, x4 + 4y4 = (x2)2 + (2y2)2

                          = (x2 + 2y2)2 – 2.x2.2y2

                          = (x2 + 2y2)2 – 4x2y2

                          = (x2 + 2y2)2 – (2xy)2                                

                          = {(x2 + 2y2) + 2xy} {(x2 + 2y2) - 2xy)  

                          = (x2 + 2xy + 2y2) (x2 - 2xy + 2y2)  

Q. 16.         81x4 + 64y4

Solution

Here, 81x4 + 64y4 = (9x2)2 + (8y2)2

                                  = (3x2 + 8y2)2 – 2.9x2.8y2

                                  = (3x2 + 8y2)2 – 144x2y2

                                  = (3x2 + 8y2)2 – (12xy)2

                                  = {(3x2 + 8y2) + 12xy} {(3x2 + 8y2) - 12xy)   

                                  = (3x2 + 12xy + 8y2) (3x2 - 12xy + 8y2)   

Q.17.         x4 - 7x2 + 1

Solution

Here, x4 - 7x2 + 1 = x4 + 1 - 7x2

                                  = (x2)2 + (1)2 - 7x2

                                  = (x2 + 1)2 - 2.x2.1 - 7x2

                                  = (x2 + 1)2 - 9x2

                                  = (x2 + 1)2 - (3x) 2

                                  = {(x2 + 1) + 3x}{(x2 + 1)- 3x}

                                  = (x2 + 3x + 1)(x2 - 3x + 1)

 

Q. 18.         x4 - 5x2y2 + 4y4

Solution

Here, x4 - 5x2y2 + 4y4 = x4  + 4y4 - 5x2y2

                                          = (x2)2 + (2y2)2 - 5x2y2

                                          = (x2 + 2y2)2 - 2.x2.2y2 - 5x2y2

                                          = (x2 + 2y2)2 - 4x2y2 - 5x2y2

                                          = (x2 + 2y2)2 - 9x2y2

                                          = (x2 + 2y2)2 - (3xy)2 

                                          = {(x2 + 2y2) + 3xy} {(x2 + 2y2) - 3xy}

                                          = (x2 + 3xy + 2y2)(x2 - 3xy + 2y2)

 

Q. 19.         49x4 - 154x2y2 + 9y4

Solution

Here, 49x4 - 154x2y2 + 9y4 = 49x4 + 9y4 - 154x2y2

                                                    = (7x2)2 + (3y2)2 - 154x2y2

                                                    = (7x2 + 3y2)2 - 2.7x2.3y2- 154x2y2

                                                    = (7x2 + 3y2)2 - 196x2y2



23.       x4 - 8x2 - 33 - 14y - y2

Solution: x4 - 8x2 - 33 - 14y - y2 = (x2)2 - 2.x2.4 + (4)2 - (4)2 - 33 - 14y - y2

                                                 = (x2 - 4)2 - 16 - 33 - 14y + y2                                                                                             

                                            = (x2 - 4)2 - 49 - 14y - y2                                                                                                       

                                            = (x2 - 4)2 - (49 + 14y + y2)                                                                                                     

                                            = (x2 - 4)2 - {(7)2 + 2.7.y + y2)}

                                            = (x2 - 4)2 - (7 + y)2

                                            = {(x2 - 4) + (7 + y)} {(x2 - 4) - (7 + y)}

                                            = (x2 - 4 + 7 + y) (x2 - 4 - 7 - y)

                                            = (x2  + 3 + y) (x2 - 11 - y)

24.       x4 - 6x2 - 7 - 8x - x2

Solution

Here, x4 - 6x2 - 7 - 8x - x2

                  = (x2)2 - 2.x2.3 + (3)2 - (3)2 - 7 - 8x - x2

= (x2 - 3)2 - 9 - 7 - 8x - x2

= (x2 - 3)2 - 16 - 8x - x2

 = (x2 - 3)2 - (16 + 8x + x2)

        = (x2 - 3)2 - {(4)2 + 2.4.x + x2}

= (x2 - 3)2 - (4 + x)2

             = {(x2 - 3) + (4 + x)} {(x2 - 3) - (4 + x)}

             = (x2 - 3 + 4 + x) (x2 - 3 - 4 - x)

             = (x2 + x + 1) (x2 - x - 7) Ans

25.       x4 - 12x2 - 28 + 16y - y2

Solution

Here, x4 - 12x2 - 28 + 16y - y2

= (x2)2 - 2.x2.6 + (6)2 - (6)2 - 28 + 16y - y2

 = (x2 - 6)2 -36 - 28 + 16y - y2

 = (x2 - 6)2 - 64 + 16y - y2

= (x2 - 6)2 - (64 - 16y + y2)

= (x2 - 6)2 - {(8)2 - 2.8.y + y2}

26.       x4 - 10x2 + 24 + 6y2 - 9y4

Solution

Here, x4 - 10x2 + 24 + 6y2 - 9y4

             = (x2)2- 2.x2.5 + (5)2 - (5)2 + 24 + 6y2- 9y4

             = (x2 - 5)2 - 25 + 24 + 6y2- 9y4

             = (x2 - 5)2 - 1 + 6y2- 9y4

             = (x2 - 5)2 - (1 - 6y2- 9y4)

             = (x2 - 5)2 - {(1)2 - 2.1.3y2+ (3y2)2}

             = (x2 - 5)2 - (1 - 3y2)2

 

27.       x2- 10xy + 16y2 - z2 + 6yz

Solution

Here, x2 - 10xy + 16y2 - z2 + 6yz

             = x2 - 2.x.5y + (5y)2 - (5y)2 + 16y2 - z2 + 6yz

             = (x2- 5y)2 - 25y2 + 16y2 - z2 + 6yz

             = (x2 - 5y)2 - 9y2 - z2 + 6yz

             = (x2- 5y)2 - (9y2+ z2 - 6yz)

             = (x2- 5y)2 - (9y2 - 6yz+ z2)

             = (x2- 5y)2 - {(3y)2 - 2. 3y . + z2)

 

28.       4225x4- 130x2 - 3 + 36y2 - 81y4

Solution

Here, 4225x4- 130x2 - 3 + 36y2 - 81y4

             = (65x2)2 - 2.65x2. 1 + (1)2 - (1)2 - 3 + 36y2 - 81y4

             = (65x2 - 1)2 - 1 - 3 + 36y2 - 81y4

             = (65x2 - 1)2 - 4 + 36y2 - 81y4

             = (65x2 - 1)2 - (4 - 36y2 + 81y4)

             = (65x2 - 1)2 - {(2)2 - 2.2.9y2 + (9y2)2}

             = (65x2 - 1)2 - (2 - 9y2)2

             = {(65x2 - 1) + (2 - 9y2)} {(65x2 - 1) - (2 - 9y2)}

             = (65x2 - 1 + 2 - 9y2) (65x2 - 1 - 2 + 9y2)

             = (65x2 + 1 - 9y2) (65x2 - 3 + 9y2) Ans

29.       x2 - 90xy + 2050y2 - 550yz + 3025z2

Solution

Here, x2 - 90xy + 2050y2 - 550yz + 3025z2

             = x2 - 2.x.45y + (45y)2 - (45y)2 - 2050y2 - 550yz - 3025z2

             = (x - 45y)2 - 2025y2 + 2050y2 - 550yz - 3025z2

             = (x - 45y)2 + 25y2 - 550yz +  3025z2

             = (x - 45y)2 - {(5y)2 - 2 .5y . 55yz + (55z)2}

             = (x - 45y)2 - (5y - 55z)2

             = {(x - 45y) + (5y - 55z)} {(x - 45y) - (5y - 55z)}

             = (x - 45y + 5y - 55z)(x - 45y - 5y + 55z)

             = (x - 40y - 55z)(x - 50y + 55z)

30.       169x2 - 52x - 56y - 196y2

Solution

Here, 169x2 - 52x - 56y - 196y2

             = (13x)2 - 2.13x .2 + (2)2 - (2)2 - 52x - 56y - 196y2

             = (13x - 2)2-  4  - 56y - 196y2

             = (13x - 2)2-  (4  + 56y + 196y2)

             = (13x - 2)2-  {(2)2  + 2.2 14y + (14y)2}

             = (13x - 2)2-  (2 + 14y)2

             = Complete it

31.  4225x2 - 130xy - 3y2 - z2 - 4yz

Solution

Here, 4225x2 - 130xy - 3y2 - z2 - 4yz

             = (65x)2 - 2. 65x. y + y2 - y2 - 3y2 - z2 - 4yz

             = (65x - y)2  - 4y2 - z2 - 4yz

             = (65x - y)2  - (4y2 + z2 + 4yz)

             = (65x - y)2  - {(2y)2 + 2.2y.z + z2)}

             = (65x - y)2  - (2y + z)2

             = Complete it

 

 

32.     289x2 + 170x + 24 - 38y2 -361y4

Solution

Here, 289x2 + 170x + 24 - 38y2 -361y4

          = (17x)2 + 2.17x.5 + (5)2 - (5)2 + 24 - 38y2 -361y4

          = (17x + 5)2 - 25 + 24 - 38y2 -361y4

          = (17x + 5)2 - 1 - 38y2 -361y4

          = (17x + 5)2 - (1 + 38y2 +361y4)

          = (17x + 5)2 - {(1)2 + 2.1.19y2 +(19y2)2}

          = (17x + 5)2 - (1 + 19y2)2

          = Complete it

33.     x2 + 50xy + 641y2 - 8yz -z2

Solution

Here, x2 + 50xy + 641y2 - 8yz -z2

          = x2 + 2.x.25y + (25y)2 - (25y)2 + 641y2 - 8yz -z2

          = (x + 25y)2 - 625y2 + 641y2 - 8yz -z2

          = (x + 25y)2 - 16y2 - 8yz -z2

          = (x + 25y)2 - (16y2 + 8yz + z2)

          = (x + 25y)2 - {(4y)2 + 2.4y.z + z2)}

          = (x + 25y)2 - (4y + z)2

          = Complete it


34.