SELINA Solution Class 9 Factorisation Chapter 5 Exercise 5C

Question 1

Factorise : 25a² - 9b²

Sol:

25a² - 9b² 
= ( 5a )2 - ( 3b )2
= ( 5a - 3b )( 5a + 3b )

Question 2

Factorise : a² - (2a + 3b)²

Sol:

 a² - (2a + 3b)² 
= a² - ( 2a + 3b )²
= ( a - 2a - 3b )( a + 2a + 3b )        [ ∵ a2 - b2 = ( a + b )(  a - b )]
= ( - a - 3b )( 3a + 3b )
= -3( a + 3b )( a + b )

Question 3

Factorise : a² - 81 (b-c)²

Sol:

 a² - 81 (b-c)² 
= ( a )² - [ 9( b - c ) ]²
= [ a - ( 9b - 9c )][ a + ( 9b - 9c )]         [ ∵ a2 - b2 = ( a + b )( a - b )]
= ( a - 9b + 9c )( a + 9b - 9c )

Question 4

Factorise : 25(2a - b)² - 81b²

Sol:

25(2a - b)² - 81b²
= [ 5( 2a - b )]² - (9b)²
= [ 5( 2a - b ) - 9b ][ 5( 2a - b ) + 9b ]               
                                          [ ∵ a² - b² = ( a + b )( a - b )]
= [ 10a - 5b - 9b ][ 10a - 5b + 9b ]
= [ 10a - 14b ][ 10a + 4b ]
= 2 x ( 5a - 7b ) x 2 x ( 5a + 2b )
= 4( 5a - 7b )( 5a + 2b )

Question 5

Factorise : 50a³ - 2a

Sol:

50a³ - 2a 
= 2a( 25a² - 1 )
= 2a[ (5a)² - (1)² ]
= 2a( 5a + 1 )( 5a - 1 )      [ ∵ a² - b² = ( a + b )( a - b )]

Question 6

Factorise : 4a²b - 9b³

Sol:

4a²b - 9b³
= b( 4a² - 9b² )
= b[ (2a)² - (3b)² ]
= b( 2a - 3b )( 2a + 3b )   [ ∵ a2 - b2 = ( a + b )( a - b )]

Question 7

Factorise : 3a⁵ - 108a³

Sol:

3a⁵ - 108a³ 
= 3a³( a² - 36 )
= 3a³[( a )² - ( 6 )²]
= 3a³( a - 6 )( a + 6 )         [ ∵ a² - b² = ( a + b )( a - b )]

Question 8

Factorise : 9(a - 2)² - 16(a + 2)²

Sol:

 9(a - 2)² - 16(a + 2)²
= [ 3( a - 2 )]² - [4( a + 2 )]²
= [ 3( a - 2 ) - 4( a + 2 )][ 3( a - 2) + 4( a + 2 )]
                                   [ ∵ a² - b² = ( a + b )( a - b )]
= [ 3a - 6 - 4a - 8 ][ 3a - 6 + 4a + 8 ]
= ( - a - 14 )( 7a + 2 )
= - ( a + 14 )( 7a + 2 )

Question 9

Factorise : a⁴ - 1

Sol:

a⁴ - 1
= ( a² )² - ( 1 )²
= ( a² + 1 )( a² - 1 )        [ ∵ a² - b² = ( a + b )( a - b )]
= ( a² + 1 )[ (a)² - (1)² ]
= ( a² + 1 )( a + 1 )( a - 1 )       

Question 10

Factorise : a³ + 2a² - a - 2

Sol:

a³ + 2a² - a - 2
= a²( a + 2 ) - 1( a + 2 )
= ( a² - 1 )( a + 2 )
= ( a + 1 )( a - 1 )( a + 2 )   [ ∵ a² - b² = ( a + b )( a - b )] 

Question 11

Factorise : (a + b)³ - a - b

Sol:

(a + b)³ - a - b 
= ( a + b )³ - ( a + b )
= ( a + b )[ ( a + b )² - 1 ]
= ( a + b )[ ( a + b )² - (1)² ]
= ( a + b )[( a + b ) + 1 ][( a + b ) - 1]             [ ∵ a² - b² = ( a + b )( a - b )] 
= ( a + b )( a + b + 1 )( a + b - 1 )

Question 12

Factorise : a (a - 1) - b (b - 1)

Sol:

a (a - 1) - b (b - 1)
= a² - a - b² + b
= a² - b² - a + b
= ( a + b )( a - b ) - ( a - b )  [∵ a² - b² = ( a + b )( a - b )]
= ( a - b )[( a + b ) - 1]
= ( a - b )[ a + b - 1 ]

Question 13

Factorise : 4a² - (4b² + 4bc + c²)

Sol:

4a² - (4b² + 4bc + c²)
 = ( 2a )² - ( 2b + c )²
= [ 2a - ( 2b + c )][ 2a + (2b + c )]      [ ∵ a² - b² = ( a + b )( a - b )]
= [ 2a - 2b - c ][ 2a + 2b + c ]

Question 14

Factorise : 4a² - 49b² + 2a - 7b

Sol:

4a² - 49b² + 2a - 7b 
= [ ( 2a )² - ( 7b )²] + [ 2a - 7b ]
= [ 2a - 7b ][ 2a + 7b ] + [ 2a - 7b ]       [ ∵ a² - b² = ( a + b )( a - b ) ]
= [ 2a - 7b ][ 2a + 7b + 1 ]

Question 15

Factorise : 9a² + 3a - 8b - 64b²

Sol:

9a² + 3a - 8b - 64b²
= 9a²  - 64b² + 3a - 8b
= ( 3a )² - ( 8b )² + 3a - 8b
= ( 3a - 8b )( 3a + 8b ) + ( 3a - 8b ) 
                                     [ ∵ a² - b² = ( a + b )( a - b )]
= ( 3a - 8b )( 3a + 8b + 1 )

Question 16

Factorise : 4a² - 12a + 9 - 49b²

Sol:

4a² - 12a + 9 - 49b² 
= ( 2a )2 - 12a + (3)2 - 49b2
= (2a - 3)2 - 49b2
= ( 2a - 3)2 - (7b)2
= ( 2a - 3 - 7b )( 2a - 3 + 7b )            [ ∵ a2 - b2 = ( a + b )( a - b )]

Question 17

Factorise : 4xy - x² - 4y² + z²

Sol:

4xy - x² - 4y² + z²
= z² - ( x² + 4y² - 4xy )
= z² - ( x - 2y )²
= [ z - ( x - 2y )][ z + ( x - 2y )]             [ ∵ a² - b² = ( a + b )( a - b )]
= [ z - x + 2y ][ z + x - 2y ]

Question 18

Factorise : a² + b² - c² - d² + 2ab - 2cd

Sol:

a² + b² - c² - d² + 2ab - 2cd
= ( a² + b² + 2ab ) - ( c² + d² + 2cd )
= ( a + b )² - ( c + d )²
= [( a + b ) - ( c + d )][( a + b ) + ( c + d )]   [∵ a² - b² = ( a + b )( a - b )]
= ( a + b - c - d )( a + b + c + d )

Question 19

Factorise : 4x² - 12ax - y² - z² - 2yz + 9a²

Sol:

4x² - 12ax - y² - z² - 2yz + 9a² 
= 4x² + 9a² - 12ax - y² - z² - 2yz
= ( 2x )² + ( 3a )² - 12ax - ( y² + z² + 2yz )
= ( 2x - 3a )² - ( y + z )²
= [( 2x - 3a ) - ( y + z )][( 2x - 3a ) + ( y + z )]     
[ ∵ a² - b² = ( a + b )( a - b )]
= [ 2x - 3a - y - z ][ 2x - 3a + y + z ]

Question 20

Factorise : (a² - 1) (b² - 1) + 4ab

Sol:

(a² - 1) (b² - 1) + 4ab
= a²b² - a² - b² + 1 + 4ab
= a²b² + 1 + 2ab - a² - b² + 2ab
= ( a²b² + 1 + 2ab ) - ( a² + b² - 2ab )
= ( ab + 1)² - ( a - b )²
= [( ab + 1 ) - ( a - b )][( ab + 1 ) + ( a - b )]                 [ ∵ a² - b² = ( a + b )( a - b )]
= [ ab + 1 - a + b ][ ab + 1 + a - b ]

Question 21

Factorise : x⁴ + x² + 1

Sol:

x⁴ + x² + 1
= x⁴ + 2x² + 1 - x²
= (x²)² + 2x² + (1)² - x²
= ( x² + 1 )² - (x)²             [ ∵ a² - b² = ( a + b )( a - b )]
= ( x² + 1 - x )( x² + 1 + x )

Question 22

Factorise : (a² + b² - 4c²)² - 4a²b²

Sol:

(a² + b² - 4c²)² - 4a²b²
= ( a² + b² - 4c² )² - ( 2ab )²
= ( a² + b² - 4c² - 2ab )( a² + b² - 4c² + 2ab )         [ ∵ a² - b² = ( a + b )( a - b )]
= ( a² + b² - 2ab - 4c² )( a² + b² + 2ab - 4c² )
= [ ( a - b )² - ( 2c )² ][ ( a + b )² - ( 2c )²]
= ( a - b + 2c )( a - b - 2c )( a + b + 2c )( a + b - 2c )

Question 23

Factorise : (x² + 4y² - 9z²)² - 16x²y²

Sol:

(x² + 4y² - 9z²)² - 16x²y²

= (x² + 4y² - 9z²)² - ( 4xy )²

= ( x² + 4y² - 9z² - 4xy )( x² + 4y² - 9z² + 4xy )      [ ∵ a² - b² = ( a + b )( a - b )]

= ( x² + 4y² - 4xy - 9z² )( x² + 4y² + 4xy - 9z² )

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

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

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

Question 24

Factorise : (a + b) ² - a² + b²

Sol:

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

Question 25

Factorise : a² - b² - (a + b) ²

Sol:

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

Question 26

Factorize : 9a² - (a² - 4) ²

Sol:

9a² - (a² - 4)² 
= (3a)² - (a² - 4)²
= [ 3a + (a² - 4)][ 3a - (a² - 4)]
= [ 3a + a² - 4 ][ 3a - a² + 4 ] 
= [ a² + 3a - 4 ][- a² + 3a + 4]
= (a + 4)(a - 1)(-a² + 3a + 4)
= (a + 4)(a - 1)[-(a² - 3a - 4)]
= - (a + 4)(a - 1)(a - 4)(a + 1)
= (a + 4)(a - 1)(a + 1)(4 - a).

Question 27

Factorise : ${\displaystyle x^2+\frac{1}{x^2}-11}$

Sol:

${\displaystyle x^2+\frac{1}{x^2}-11}$

= ${\displaystyle x^2+\frac{1}{x^2}-2-9}$

= ${\displaystyle x^2+\frac{1}{x^2}-2\times x\times \frac{1}{x}-9}$

= ${\displaystyle {(x-\frac{1}{x})}^2-{\left(3\right)}^2}$

= ${\displaystyle (x-\frac{1}{x}+3)(x-\frac{1}{x}-3)}$

Question 28

Factorise : ${\displaystyle 4x^2+\frac{1}{{\left(4x\right)}^2}+1}$

Sol:

${\displaystyle 4x^2+\frac{1}{{\left(4x\right)}^2}+1}$

= ${\displaystyle 4x^2+\frac{1}{{\left(4x\right)}^2}+2-1}$

= ${\displaystyle 4x^2+\frac{1}{{\left(4x\right)}^2}+2\times 2x\times \frac{1}{2x}-1}$

= ${\displaystyle {(2x+\frac{1}{2x}+1)}^2-{\left(1\right)}^2}$

= ${\displaystyle (2x+\frac{1}{2x}+1)(2x+\frac{1}{2x}-1)}$

Question 29

Factorise : 4x⁴ - x² - 12x - 36

Sol:

Factorise : 4x⁴ - x² - 12x - 36
= 4x⁴ - ( x² + 12x + 36 )
= ( 2x²)² - ( x² + 2 x x x 6 + 6² )
= ( 2x²)² - ( x + 6 )²
= ( 2x² + x + 6 )( 2x² - x - 6 )
= ( 2x² + x + 6 )( 2x² - 4x + 3x - 6 )
= ( 2x² + x + 6 )[ 2x( x - 2 ) + 3( x - 2 )]
= ( 2x² + x + 6 )[ ( x - 2)( 2x + 3 )]
= ( 2x² + x + 6 )( x - 2 )( 2x + 3 )

Question 30

Factorise : a² ( b + c) - (b + c)³

Sol:

a² ( b + c) - (b + c)³
= ( b + c )[ a2 - ( b + c )2 ]
= ( b + c )[( a + b + c )( a - b - c )]
= ( b + c )( a + b + c )( a - b - c )