Chapter 5 Factorization Of Algebraic Expressions | RD Sharma Solution | - GMS - Learning Simply
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# Chapter 5 Factorization Of Algebraic Expressions | RD Sharma Solution |

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# RD Sharma Solutions for Class 9 Maths Chapter 5 Factorization of Algebraic Expressions

RD Sharma Class 9 Solutions for the Chapter 5 Factorization of Algebraic Expressions are given here in detail. This study material includes five exercises and all the questions are solved by GMS. RD Sharma Class 9 Solutions Maths helps to build the basics and in-depth understanding of the fundamental Maths concepts. In this chapter, students will learn about factorisation and how to simplify an algebraic expression using the factorisation method. The process of factorization can be defined as the disintegration of a term into smaller factors. Whereas, the algebraic expressions are built up of variables, integer constants, and basic arithmetic operations of algebra.

### Exercise 5.1 Page No: 5.9

Question 1: Factorize x3 + x – 3x2 – 3

Solution:

x3 + x – 3x2 – 3

Here x is common factor in x3 + x and – 3 is common factor in – 3x2 – 3

x3 – 3x2 + x – 3

x2 (x – 3) + 1(x – 3)

Taking ( x – 3) common

(x – 3) (x2 + 1)

Therefore x3 + x – 3x2 – 3 = (x – 3) (x2 + 1)

Question 2: Factorize a(a + b)3 – 3a2b(a + b)

Solution:

a(a + b)3 – 3a2b(a + b)

Taking (a + b) as common factor

= a(a + b) {(a + b)2 – 3ab}

= a(a + b) {a2 + b2 + 2ab – 3ab}

= a(a + b) (a2 + b2 – ab)

Question 3: Factorize x(x3 – y3) + 3xy(x – y)

Solution:

x(x3 – y3) + 3xy(x – y)

= x(x – y) (x2 + xy + y2) + 3xy(x – y)

Taking x(x – y) as a common factor

= x(x – y) (x2 + xy + y2 + 3y)

= x(x – y) (x2 + xy + y2 + 3y)

Question 4: Factorize a2x2 + (ax2 + 1)x + a

Solution:

a2x2 + (ax2 + 1)x + a

= a2x2 + a + (ax2 + 1)x

= a(ax2 + 1) + x(ax2 + 1)

= (ax2 + 1) (a + x)

Question 5: Factorize x2 + y – xy – x

Solution:

x2 + y – xy – x

= x– x – xy + y

= x(x- 1) – y(x – 1)

= (x – 1) (x – y)

Question 6: Factorize x3 – 2x2y + 3xy2 – 6y3

Solution:

x3 – 2x2y + 3xy2 – 6y3

= x2(x – 2y) + 3y2(x – 2y)

= (x – 2y) (x2 + 3y2)

Question 7: Factorize 6ab – b2 + 12ac – 2bc

Solution:

6ab – b2 + 12ac – 2bc

= 6ab + 12ac – b2 – 2bc

Taking 6a common from first two terms and –b from last two terms

= 6a(b + 2c) – b(b + 2c)

Taking (b + 2c) common factor

= (b + 2c) (6a – b)

Question 8: Factorize (x2 + 1/x2) – 4(x + 1/x) + 6

Solution:

(x2 + 1/x2) – 4(x + 1/x) + 6

= x2 + 1/x2 – 4x – 4/x + 4 + 2

= x2 + 1/x2 + 4 + 2 – 4/x – 4x

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

As we know, x2 + y2 + z2 + 2xy + 2yz + 2zx = (x+y+z) 2

So, we can write;

= (x + 1/x + (-2 )) 2

or (x + 1/x – 2) 2

Therefore, x2 + 1/x2) – 4(x + 1/x) + 6 = (x + 1/x – 2) 2

Question 9: Factorize x(x – 2) (x – 4) + 4x – 8

Solution:

x(x – 2) (x – 4) + 4x – 8

= x(x – 2) (x – 4) + 4(x – 2)

= (x – 2) [x(x – 4) + 4]

= (x – 2) (x2 – 4x + 4)

= (x – 2) [x2 – 2 (x)(2) + (2) 2]

= (x – 2) (x – 2) 2

= (x – 2)3

Question 10: Factorize ( x + 2 ) ( x2 + 25 ) – 10x2 – 20x

Solution :

( x + 2) ( x2 + 25) – 10x ( x + 2 )

Take ( x + 2 ) as common factor;

= ( x + 2 )( x+ 25 – 10x)

=( x + 2 ) ( x– 10x + 25)

Expanding the middle term of ( x2 – 10x + 25 )

=( x + 2 ) ( x– 5x – 5x + 25 )

=( x + 2 ){ x (x – 5 ) – 5 ( x – 5 )}

=( x + 2 )( x – 5 )( x – 5 )

=( x + 2 )( x – 5 )2

Therefore, ( x + 2) ( x2 + 25) – 10x ( x + 2 ) = ( x + 2 )( x – 5 )2

Question 11: Factorize 2a2 + 2√6 ab + 3b2

Solution:

2a2 + 2√6 ab + 3b2

Above expression can be written as ( √2a )2 + 2 × √2a × √3b + ( √3b)2

As we know, ( p + q ) 2 = p+ q2 + 2pq

Here p = √2a and q = √3b

= (√2a + √3b )2

Therefore, 2a2 + 2√6 ab + 3b2 = (√2a + √3b )2

Question 12: Factorize (a – b + c)2 + (b – c + a) 2 + 2(a – b + c) (b – c + a)

Solution:

(a – b + c)2 + ( b – c + a) 2 + 2(a – b + c) (b – c + a)

{Because p+ q2 + 2pq = (p + q) 2}

Here p = a – b + c and q = b – c + a

= [a – b + c + b- c + a]2

= (2a)2

= 4a2

Question 13: Factorize a2 + b+ 2( ab+bc+ca )

Solution:

a2 + b+ 2ab + 2bc + 2ca

As we know, p+ q2 + 2pq = (p + q) 2

We get,

= ( a+b)2 + 2bc + 2ca

= ( a+b)2 + 2c( b + a )

Or ( a+b)2 + 2c( a + b )

Take ( a + b ) as common factor;

= ( a + b )( a + b + 2c )

Therefore, a2 + b+ 2ab + 2bc + 2ca = ( a + b )( a + b + 2c )

Question 14: Factorize 4(x-y) 2 – 12(x – y)(x + y) + 9(x + y)2

Solution :

Consider ( x – y ) = p, ( x + y ) = q

= 4p2 – 12pq + 9q2

Expanding the middle term, -12 = -6 -6 also 4× 9=-6 × -6

= 4p2 – 6pq – 6pq + 9q2

=2p( 2p – 3q ) -3q( 2p – 3q )

= ( 2p – 3q ) ( 2p – 3q )

= ( 2p – 3q )2

Substituting back p = x – y and q = x + y;

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

= (2x-3x-2y-3y ) 2

=[ -x – 5y] 2

=[( -1 )( x+5y )] 2

=( x+5y ) 2

Therefore, 4(x-y) 2 – 12(x – y)(x + y) + 9(x + y)2 = ( x+5y )2

Question 15: Factorize a2 – b+ 2bc – c2

Solution :

a2 – b+ 2bc – c2

As we know, ( a-b)2 = a+ b– 2ab

= a– ( b – c) 2

Also we know, a– b= ( a+b)( a-b)

= ( a + b – c )( a – ( b – c ))

= ( a + b – c )( a – b + c )

Therefore, a2 – b+ 2bc – c=( a + b – c )( a – b + c )

Question 16: Factorize a2 + 2ab + b2 – c2

Solution:

a2 + 2ab + b2 – c2

= (a2 + 2ab + b2) – c2

= (a + b)2 – (c) 2

We know, a2 – b2 = (a + b) (a – b)

= (a + b + c) (a + b – c)

Therefore a2 + 2ab + b2 – c= (a + b + c) (a + b – c)

### Exercise 5.2 Page No: 5.13

Factorize each of the following expressions:

Question 1: p3 + 27

Solution:

p3 + 27

= p3 + 33

[using a3 + b= (a + b)(a2 –ab + b2)]

= (p + 3)(p² – 3p – 9)

Therefore, p3 + 27 = (p + 3)(p² – 3p – 9)

Question 2: y3 + 125

Solution:

y3 + 125

= y3 + 53

[using a3 + b= (a + b)(a2 –ab + b2)]

= (y+5)(y2 − 5y + 52)

= (y + 5)(y− 5y + 25)

Therefore, y3 + 125 = (y + 5)(y− 5y + 25)

Question 3: 1 – 27a3

Solution:

= (1)3 −(3a) 3

[using a3 – b= (a – b)(a2 + ab + b2)]

= (1− 3a)(12 + 1×3a + (3a) 2)

= (1−3a)(1 + 3a + 9a2)

Therefore, 1−27a3 = (1−3a)(1 + 3a+ 9a2)

Question 4: 8x3y3 + 27a3

Solution:

8x3y3 + 27a3

= (2xy) 3 + (3a) 3

[using a3 + b= (a + b)(a2 –ab + b2)]

= (2xy +3a)((2xy)2−2xy×3a+(3a) 2)

= (2xy+3a)(4x2y−6xya + 9a2)

Question 5: 64a3 − b3

Solution:

64a3 − b3

= (4a)3−b3

[using a3 – b= (a – b)(a2 + ab + b2)]

= (4a−b)((4a)2 + 4a×b + b2)

=(4a−b)(16a+4ab+b2)

Question 6: x3 / 216 – 8y3

Solution:

x3 / 216 – 8y3

Question 7: 10xy – 10xy4

Solution:

10xy – 10xy4

= 10xy(x− y3)

[using a3 – b= (a – b)(a2 + ab + b2)]

= 10xy (x−y)(x+ xy + y2)

Therefore, 10xy – 10xy= 10xy (x−y)(x+ xy + y2)

Question 8: 54xy + 2x3y4

Solution:

54xy + 2x3y4

= 2x3y(27x3 +y3)

= 2x3y((3x) 3 + y3)

[using a3 + b= (a + b)(a2 – ab + b2)]

= 2x3y {(3x+y) ((3x)2−3xy+y2)}

=2x3y(3x+y)(9x2 − 3xy + y2)

Question 9: 32a+ 108b3

Solution:

32a+ 108b3

= 4(8a3 + 27b3)

= 4((2a) 3+(3b) 3)

[using a3 + b= (a + b)(a2 – ab + b2)]

= 4[(2a+3b)((2a)2−2a×3b+(3b) 2)]

= 4(2a+3b)(4a− 6ab + 9b2)

Question 10: (a−2b)3 − 512b3

Solution:

(a−2b)3 − 512b3

= (a−2b)−(8b) 3

[using a3 – b= (a – b)(a2 + ab + b2)]

= (a −2b−8b) {(a−2b)+ (a−2b)8b + (8b) 2}

=(a −10b)(a+ 4b− 4ab + 8ab − 16b+ 64b2)

=(a−10b)(a+ 52b+ 4ab)

Question 11: (a+b)3 − 8(a−b)3

Solution:

(a+b)3 − 8(a−b)3

= (a+b)− [2(a−b)]3

= (a+b)3 − [2a−2b] 3

[using p3 – q= (p – q)(p2 + pq + q2)]

Here p = a+b and q = 2a−2b

= (a+b−(2a−2b))((a+b)2+(a+b)(2a−2b)+(2a−2b) 2)

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

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

=(3b−a)(3a2+2ab−b2+(2a−2b) 2)

=(3b−a)(3a2+2ab−b2+4a2+4b2−8ab)

=(3b−a)(3a2+4a2−b2+4b2−8ab+2ab)

=(3b−a)(7a2+3b2−6ab)

Question 12: (x+2)+ (x−2) 3

Solution:

(x+2)+ (x−2) 3

[using p3 + q= (p + q)(p2 – pq + q2)]

Here p = x + 2 and q = x – 2

= (x+2+x−2)((x+2)2−(x+2)(x−2)+(x−2) 2)

=2x(x+4x+4−(x+2)(x−2)+x2−4x+4)

[ Using : (a+b)(a−b) = a2−b2 ]

= 2x(2x+ 8 − (x− 22))

= 2x(2x+8 − x+ 4)

= 2x(x+ 12)

### Exercise 5.3 Page No: 5.17

Question 1: Factorize 64a3 + 125b3 + 240a2b + 300ab2

Solution:

64a3 + 125b3 + 240a2b + 300ab2

= (4a)3 + (5b) 3 + 3(4a)2(5b) + 3(4a)(5b)2 , which is similar to a3 + b+ 3a2b + 3ab2

We know that, a3 + b+ 3a2b + 3ab= (a+b)3]

= (4a+5b)3

Question 2: Factorize 125x3 – 27y3 – 225x2y + 135xy2

Solution:

125x3 – 27y3 – 225x2y + 135xy2

Above expression can be written as (5x)3−(3y) 3−3(5x)2(3y) + 3(5x)(3y)2

Using: a3 − b3 − 3a2b + 3ab2 = (a−b)3

= (5x − 3y)3

Question 3: Factorize 8/27 x3 + 1 + 4/3 x2 + 2x

Solution:

8/27 x3 + 1 + 4/3 x2 + 2x

Question 4: Factorize 8x3 + 27y3 + 36x2y + 54xy2

Solution:

8x3 + 27y3 + 36x2y + 54xy2

Above expression can be written as (2x)3 + (3y) 3 + 3×(2x)2×3y + 3×(2x)(3y)2

Which is similar to a³ + b³ + 3a²b + 3ab² = (a + b) ³]

Here a = 2x and b = 3y

= (2x+3y)3

Therefore, 8x3 + 27y3 + 36x2y + 54xy2 = (2x+3y)3

Question 5: Factorize a3 − 3a2b + 3ab2 − b3 + 8

Solution:

a3 − 3a2b + 3ab2 − b3 + 8

Using: a3 − b3 − 3a2b + 3ab2 = (a−b)3

= (a−b)3 + 23

Again , Using: a3 + b3 =(a + b)(a2 – ab + b2)]

=(a−b+2)((a−b)2−(a−b) × 2 + 22)

=(a−b+2)(a2+b2−2ab−2(a−b)+4)

=(a−b+2)(a2+b2−2ab−2a+2b+4)

a3 − 3a2b + 3ab2 − b3 + 8 =(a−b+2)(a2+b2−2ab−2a+2b+4)

### Exercise 5.4 Page No: 5.22

Factorize each of the following expressions:

Question 1: a3 + 8b+ 64c− 24abc

Solution:

a3 + 8b+ 64c− 24abc

= (a)3 + (2b) 3 + (4c) 3− 3×a×2b×4c

[Using a3+b3+c3−3abc = (a+b+c)(a2+b2+c2−ab−bc−ca)]

= (a+2b+4c)(a2+(2b)2 + (4c)2−a×2b−2b×4c−4c×a)

= (a+2b+4c)(a+4b+16c−2ab−8bc−4ac)

Therefore, a3 + 8b+ 64c− 24abc = (a+2b+4c)(a+4b+16c−2ab−8bc−4ac)

Question 2: x 3 − 8y 3+ 27z3 + 18xyz

Solution:

= x3 − (2y) 3 + (3z) 3 − 3×x×(−2y)(3z)

= (x + (−2y) + 3z) (x2 + (−2y)+ (3z) 2 −x(−2y)−(−2y)(3z)−3z(x))

[using a3+b3+c3−3abc = (a+b+c)(a2+b2+c2−ab−bc−ca)]

=(x −2y + 3z)(x2 + 4y+ 9 z+ 2xy + 6yz − 3zx)

Question 3: 27x 3 − y 3– z3 – 9xyz

Solution:

27x 3 − y 3– z3 – 9xyz

= (3x) 3 − y 3– z3 – 3(3xyz)

[Using a3 + b3 + c−3abc = (a + b + c)(a2+b2+c2−ab−bc−ca)]

Here a = 3x, b = -y and c = -z

= (3x – y – z){ (3x)2 + (- y)2 + (– z)2 + 3xy – yz + 3xz)}

= (3x – y – z){ 9x2 + y2 + z2 + 3xy – yz + 3xz)}

Question 4: 1/27 x3 − y3 + 125z3 + 5xyz

Solution:

1/27 x3 − y3 + 125z3 + 5xyz

= (x/3)3+(−y)3 +(5z)3 – 3 x/3 (−y)(5z)

[Using a3 + b3 + c−3abc = (a + b + c)(a2+b2+c2−ab−bc−ca)]

= (x/3 + (−y) + 5z)((x/3)2 + (−y)2 + (5z) 2 –x/3(−y) − (−y)5z−5z(x/3))

= (x/3 −y + 5z) (x^2/9 + y2 + 25z+ xy/3 + 5yz – 5zx/3)

Question 5: 8x3 + 27y− 216z+ 108xyz

Solution:

8x3 + 27y− 216z+ 108xyz

= (2x) 3 + (3y) 3 +(−6y) 3 −3(2x)(3y)(−6z)

= (2x+3y+(−6z)){ (2x)2+(3y) 2+(−6z) 2 −2x×3y−3y(−6z)−(−6z)2x}

= (2x+3y−6z) {4x+9y+36z−6xy + 18yz + 12zx}

Question 6: 125 + 8x3 − 27y3 + 90xy

Solution:

125 + 8x3 − 27y3 + 90xy

= (5)3 + (2x) 3 +(−3y) 3 −3×5×2x×(−3y)

= (5+2x+(−3y)) (52 +(2x) 2 +(−3y) 2 −5(2x)−2x(−3y)−(−3y)5)

= (5+2x−3y)(25+4x+9y−10x+6xy+15y)

Question 7: (3x−2y)3 + (2y−4z) 3 + (4z−3x) 3

Solution:

(3x−2y)3 + (2y−4z) 3 + (4z−3x) 3

Let (3x−2y) = a, (2y−4z) = b , (4z−3x) = c

a + b + c= 3x−2y+2y−4z+4z−3x = 0

We know, a3 + b3 + c−3abc = (a + b + c)(a2+b2+c2−ab−bc−ca)

⇒ a3 + b3 + c−3abc = 0

or a3 + b3 + c=3abc

⇒ (3x−2y)3 + (2y−4z) 3 + (4z−3x) 3 = 3(3x−2y)(2y−4z)(4z−3x)

Question 8: (2x−3y)3 + (4z−2x) 3 + (3y−4z) 3

Solution:

(2x−3y)3 + (4z−2x) 3 + (3y−4z) 3

Let 2x – 3y = a , 4z – 2x = b , 3y – 4z = c

a + b + c= 2x – 3y + 4z – 2x + 3y – 4z = 0

We know, a3 + b3 + c−3abc = (a + b + c)(a2+b2+c2−ab−bc−ca)

⇒ a3 + b3 + c−3abc = 0

(2x−3y)3 + (4z−2x) 3 + (3y−4z) 3 = 3(2x−3y)(4z−2x)(3y−4z)

### Exercise VSAQs Page No: 5.24

Question 1: Factorize x4 + x2 + 25

Solution:

x4 + x2 + 25

= (x2) 2 + 5+ x2

[using a2 + b2 = (a + b) 2 – 2ab ]

= (x+5) 2 −2(x) (5) + x2

=(x+5) 2 −10x+ x2

=(x+ 5) 2 − 9x2

=(x+ 5) 2 − (3x) 2

[using a2 – b2 = (a + b)(a – b ]

= (x 2 + 3x + 5)(x− 3x + 5)

Question 2: Factorize x2 – 1 – 2a – a2

Solution:

x2 – 1 – 2a – a2

x2 – (1 + 2a + a2 )

x2 – (a + 1)2

(x – (a + 1)(x + (a + 1)

(x – a – 1)(x + a + 1)

[using a2 – b2 = (a + b)(a – b) and (a + b)^2 = a^2 + b^2 + 2ab ]

Question 3: If a + b + c =0, then write the value of a3 + b3 + c3.

Solution:

We know, a3 + b3 + c3 – 3abc = (a + b +c ) (a2 + b+ c– ab – bc − ca)

Put a + b + c =0

This implies

a3 + b3 + c3 = 3abc

Question 4: If a2 + b2 + c2 = 20 and a + b + c =0, find ab + bc + ca.

Solution:

We know, (a+b+c)² = a² + b² + c² + 2(ab + bc + ca)

0 = 20 + 2(ab + bc + ca)

-10 = ab + bc + ca

Or ab + bc + ca = -10

Question 5: If a + b + c = 9 and ab + bc + ca = 40, find a2 + b2 + c2 .

Solution:

We know, (a+b+c)² = a² + b² + c² + 2(ab + bc + ca)

92 = a² + b² + c² + 2(40)

81 = a² + b² + c² + 80

⇒ a² + b² + c² = 1

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