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 + b² - 16

Solution:

a² + 2ab + b² -  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)