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Chapter 7: Factorization Exercise – 7.6

Question: 1

Solve:

4x2 + 12xy + 9y2

Solution:

= (2x)2 + 2 x 2x x 3y + (3y)2

= (2x + 3y)2

= (2x + 3y) (2x + 3y)

 

Question: 2

Solve:

9a2 – 24ab + 16b2

Solution:

9a2 - 24ab + 16b2

= (3a)2 - 2 x 3a x 4b + (4b)2

= (3a - 4b)2

= (3a - 4b) (3a - 4b)

 

Question: 3

Solve:

p2q2 – 6qr + 9r2 = (pq)2 - 2 x pq x 3r + (3r)2

Solution:

p2q2 – 6qr + 9r2 = (pq)2 – 2 x pq x 3r + (3r)2

= (pq – 3r)2

= (pq – 3r) (pq – 3r)

 

Question: 4

Solve:  

36a2 + 36a + 9

Solution:

36a2 + 36a + 9

= 9 (4a2 + 4a + 1) = 9{(2a)2 + 2 x 2a x 1 + 12}

= 9 (2a + 1)2

= 9 (2a + 1) (2a + 1)

 

Question: 5

Solve:

a2 + 2ab + b- 16

Solution:

a2 + 2ab + b-  16

= a2 + 2 x a x b + b2 – 16

= (a + b)2 – 42

= (a + b – 4) (a + b + 4)

 

Question: 6

Solve:

9z2 - x2 + 4xy – 4y2

Solution:

9z2 – x2 + 4xy – 4y2

= 9z2 - (x2 – 4xy + 4y2)

= 9z2 - [x2 – 2x × 2y + (2y)2]

= (3z)2- (x - 2y)2

= [3z - (x - 2y)] [3z + (x - 2y)]

= (3z - x + 2y) (3x + x -2y)

= (x - 2y + 3z) (-x + 2y + 3z)

 

Question: 7

Solve:

9a4 - 24a2b2 + 16b4 - 256

Solution:

9a4 - 24a2b2 + 16b4- 256

= (9a4 - 24a2b2 + 16b4) - 256

= [(3a2)2- 2 x 3a2 x 4b2 + (4b2)2] -162

= (3a2 - 402)2 -162

= [(3a2 – 4b2) -16] [(3a2 - 42) + 16]

= (3a2 - 4b2 -16) (3a2 - 4b2 + 16)

 

Question: 8

Solve:  

16 - a6 + 4a3b3 – 4b6

Solution:

16 – a6 + 4a3b3 – 4b6

= 16 – (a6 – 4a3b3 + 4b6)

= 42 – [(a3)2 – 2 x a3 x 2b3 + (2b3)2]

= 42 – (a3 – 2b3)2

= [4 – (a3 – 2b3)] [4 + (a3 – 2b3)]

= (4 – a3 – 2b3) (4 + a3 – 2b3)

= (a3 – 2b3 + 4) (– a3 – 2b3 + 4)

 

Question: 9

Solve:

a2 – 2ab + b2 – c2

Solution:

a2 – 2ab + b2 – c2

= (a2 – 2ab + b2) – c2

= (a2 – 2 x a x b + b2) – c2

= (a – b)2 – c2

= [(a – b) – c][ (a – b) + c]

= (a – b – c) (a – b + c)

 

Question: 10

Solve:

x2 + 2x + 1- 9y2

Solution:

x2 + 2x + 1 – 9y2

= (x2 + 2x + 1) – 9y2

= (x2 + 2× x x 1 + 1) – 9y2

= (x + 1)2 – (3y)2

= [(x + 1) – 3y] [(x + 1) – 3y]

= (x + 1 – 3y) (x + 1 + 3y)

= (x + 3y + 1) (x – 3y + 1)

 

Question: 11

Solve:

a2 + 4ab + 3b2

Solution:

a2 + 4ab + 3b2

= a2 + 4ab + 4b2 – b2

= [a2 + 2 x a x 2b + (2b)2] – b2

= (a + 2b)2 – b2

= [(a + 2b) – b] [(a + 2b) + b]

= (a + 2b – b)(a + 2b + b)

= (a + b) (a + 3b)

 

Question: 12

Solve:

96 – 4x – x2

Solution:

96 - 4x – x2

= 100 - 4 – 4x – x2

= 100 - (x2 + 4x + 4)

= 100 - (x2 + 2 x x x 2 + 22)

= 102 – (x + 2)2

= [10 – (x + 2)] [10 + (x + 2)]

= (10 – x – 2)(10 + x + 2)

= (8 – x) (12 + x)

= (x + 12) (-x + 8)

 

Question: 13

Solve:

a4 + 3a2 + 4

Solution:

a4 + 3a2 + 4

= a4 + 4a2 – a2 + 4

= (a4 + 4a2 + 4) – a2

= [(a2)2 + 2 x a2 x 2 + 22] – a2

= (a2 + 2)2 – a2

= [(a2 + 2) – a][(a2 + 2) + a]

= (a2 – a + 2)(a2 + a + 2)

 

Question: 14

Solve:

4x4 + 1

Solution:

4x4 + 1

= 4x4 + 4x2 + 1 – 4x2

= [(2x2)2 + 2 x 2x2 x 1 + 1] – 4x2

= (2x2 + 1)2 – (2x)2

= [(2x2 + 1) – 2x] [(2x2 + 1) + 2x]

= (2x2 – 2x + 1)( 2x2 + 2x + 1)

 

Question: 15

Solve:

4x4 + y4

Solution:

4x4 + y4

= 4x4 + 4x2 + y4 – 4x2y2

= [(2x2)2 + 2 x 2x2 x y + (y2) 2] – (2xy)2

= (2x2 + y2)2 – (2xy)2

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

= (2x2 – 2xy + y2)( 2x2 + 2xy + y2)

 

Question: 16

Solve:

(x + 2)2 – 6(x + 2) + 9

Solution:

(x + 2)2 – 6(x + 2) + 9

= (x + 2)2 – 2 x (x + 2) x 3 + 32

= [(x + 2) – 3]2

= (x + 2 – 3)2

= (x – 1)2

= (x – 1)(x – 1)

 

Question: 17

Solve:

25 – p2 – q2 – 2pq

Solution:

25 – p2 – q2 – 2pq

= 25 – (p2 + 2pq + q2)

= 52 – (p2 + 2 x p x q + q2)

= 52 – (p + q)2

= [5 – (p + q)] [5 + (p + q)]

= (5 – p + q) (5 + p + q)

= - (p + q - 5)(p + q + 5)

 

Question: 18

Solve:

x2 + 9y2 – 6xy – 25a2

Solution:

x2 + 9y2 – 6xy – 25a2

=(x2 – 6xy + 9y2) – 25a2

= [x2 – 2 x x x 3y + (3y)2] – 25a2

= (x – 3y)2 – (5a)2

= [(x – 3y) – 5a][(x -3y) + 5a]

= (x – 3y – 5a)( x – 3y + 5a)

 

Question: 19

49 – a2 + 8ab – 16b2

Solution:

49 – a2 + 8ab – 16b2

= 49 – (a2 – 8ab + 16b2)

= 49 – [a2 – 2 x a x 4b + (4b2)]

= 72 – (a – 4b2)

= [7 – (a – 4b)][7 + (a – 4b)]

= (7 – a + 4b)( 7 + a – 4b)

= – (a – 4b – 7)(a – 4b + 7)

= – (a – 4b + 7)(a – 4b – 7)

 

Question: 20

Solve:

a2 – 8ab + 16b2 – 25c2

Solution:

a2 – 8ab + 16b2 – 25c2

= (a2 – 8ab + 16b2) – 25c2

= [a2 – 2 x a x 4b + (4b)2] – 25c2

= (a – 4b)2 – (5c)2

= [(a – 4b) – 5c] [(a – 4b)2 + 5c]

= (a – 4b – 5c) (a – 4b + 5c)

 

Question: 21

Solve:

x2 – y2 + 6y – 9

Solution:

x2 – y2 + 6y – 9

= x2 – (y2 + 6y – 9)

= x2 – (y2 – 2 x y x 3 + 32)

= x2 – (y – 3)2

= [x – (y – 3)] [x + (y – 3)]

= (x – y + 3)(x + y - 3)

 

Question: 22

Solve:

25x2 – 10x + 1 – 36y2

Solution:

25x2 – 10x + 1 – 36y2

= (25x2 – 10x + 1) – 36y2

= [(5x)2 – 2 x 5x x 1 + 1] – 36y2

= (5x – 1)2 – (6y)2

= [(5x – 1) – 6y] [(5x – 1) + 6y]

= (5x – 1 – 6y)( 5x – 1 + 6y)

= (5x – 6y – 1)( 5x + 6y – 1)

 

Question: 23

Solve:

a2 – b2 + 2bc – c2

Solution:

a2 – b2 + 2bc – c2

= a2 – (b2 – 2bc + c2)

= a2 – (b2 – 2 x b x c + c2)

= a2 – (b – c)2

= [a – (b – c)][ a + (b – c)]

= (a – b + c)(a + b – c)

 

Question: 24

Solve:

a2 + 2ab + b2 – c2

Solution:

a2 + 2ab + b2 – c2

= (a2 + 2ab + b2) – c2

= (a2 + 2 x a x b + b2) – c2

= (a + b)2 – c2

= [(a + b) – c] [(a + b) + c]

= (a + b – c) (a + b + c)

 

Question: 25

Solve:

49 – x2 – y2 + 2xy

Solution:

49 – x2 – y2 + 2xy

= 49 – (x2 + 2xy – y2)

= 72 – (x – y)2

= [7 – (x – y)] [7 + (x – y)]

= (7 – x + y)(7 + x – y)

= (x – y + 7)(y – x + 7)

 

Question: 26

Solve:

a2 + 4b2 – 4ab – 4c2

Solution:

a2 + 4b2 – 4ab – 4c2

= (a2 + 4b2 – 4ab) – 4c2

= [a2 – 2 x a x 2b + (2b)2] – 4c2

= (a – 2b)2 – (2c)2

= [(a – 2b) – 2c] [(a – 2b) + 2c]

= (a – 2b – 2c)(a – 2b + 2c)

 

Question: 27

Solve:

x2 – y2 – 4xz + 4z2

Solution:

x2 – y2 – 4xz + 4z2

= (x2 – 4xz + 4z2) – y2

= (x – 2z)2 – y2

= [(x – 2z) – y] [(x – 2z) + y]

= (x – 2z – y)(x – 2z + y)

= (x + y – 2z)(x – y – 2z)


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