Class 8 ICSE Maths Solutions Chapter 13

Class 8 ICSE Maths Solutions Chapter 13 Factorisation (Selina Concise)

Welcome to NCTB Solutions. Here with this post we are going to help 8th class students for the Solutions of Selina Class 8 ICSE Math Book, Chapter 13, Factorisation. Here students can easily find step by step solutions of all the problems for Factorisation, Exercise 13A, 13B, 13C, 13D, 13E and 13F Also here our mathematics teacher’s are solved all the problems with easily understandable methods with proper guidance so that all the students can understand easily. Here in this post students will get chapter 13 solutions. Here in this post all the solutions are based on ICSE latest Syllabus.

Factorisation Exercise 13(A) Solution :

Question no – (1) 

Solution : 

Given, 15x + 5

= 5 (3x + 1)

Question no – (2) 

Solution : 

Given, a³ – a² + a

= a(a² – a + 1)

Question no – (3)

Solution : 

Given, 3x² + 6x³

= 3x² (1 + 2x)

Question no – (4)

Solution : 

Given, 4a² – 8ab

= 4a (a – 2b)

Question no – (5)

Solution : 

Given, 2x³b² – 4x⁵ b⁴

= 2x³b² (1 – 2x²b²)

Question no – (6)

Solution : 

Given, 15x⁴ y³ – 20x³ y

= 5x³y (3xy² – 4)

Question no – (7)

Solution : 

Given, a³b – a²b²

= b(a³ – a²b – b²)

Question no – (8)

Solution : 

Given, 6x²y + 9xy² + 4y³

= y(6x² + 9xy + 4y²)

Question no – (9) 

Solution : 

Given, 17a⁶b⁸ – 34a⁴b⁶ + 51a²b⁴

= 17 a²b⁴ (a⁴b⁴ – 2a²b² + 3)

Question no – (10) 

Solution : 

Given, 3x⁵y – 27x⁴y² + 12x³y³

= 3x³y (x² – 9xy + 4y²)

Question no – (11)

Solution : 

Given, x² (a – b) – y² (a – b) + z² (a – b)

= (a – b) (x² – y² + z²)

Question no – (12)

Solution : 

x + y) (a + b) + (x – y) (a + b)

= (a + b) (x + y + x – y)

= (a + b) 2x

Question no – (13) 

Solution : 

Given, 2b (2a + b) – 3c (2a + b)

= (2a + b) (2b – 3c)

Question no – (14) 

Solution : 

Given, 12abc – 6a²b²c² + 3a³b³c³

= 3abc (4 – 2abc + a²b²c²)

Question no – (15) 

Solution : 

Given, 4x (3x – 2y) – 2y (3x – 2y)

= (3x – 2y) (4x – 2y

Question no – (16) 

Solution : 

Given, (a + 2b) (3a + b) – (a + b) (a + 2b) + (a + 2b)2

= (a + 2b) (3a + b – a – b + a + 2b)

= (a + 2b) (3a + 2b)

Question no – (17) 

Solution : 

Given, 6xy (a² + b²) + 8yz (a² + b²) – 10xz (a² + b²)

= 2(a² + b²) (3xy + 4yz – 5xz)

Factorisation Exercise 13(B) Solution :

Question no – (1) 

Solution : 

Given, a² + ax + ab + bx

= a² + ab + ax + bx

= a (a + b) + x (a + b)

= (a + b) (a + x)

Question no – (2) 

Solution : 

Given, a² – ab – ca + bc

= a (a – b) – c (a – b)

= (a – b) (a – c)

Question no – (3) 

Solution : 

Given, ab – 2b + a² – 2a

= b(a – 2) + a (a -2)

= (a – 2) ( b + a)

Question no – (4) 

Solution : 

Given, a³ – a² + a – 1

= a² (a – 1) + 1 (a -1)

= (a² + 1) (a – 1)

Question no – (5) 

Solution : 

Given, 2a – 4b – ca + 2bx

= 2a – xa – 4b + 2bx

= a (2 – x) – 2b (2 – x)

= (a – 2b) (2 – x)

Question no – (6) 

Solution : 

Given, xy – ay – ax + a² + bx – ab

= y (x – a) – a (x – a) + b (x – a)

= (x – a) (y – a + b)

Question no – (7) 

Solution : 

3x⁵ – 6x⁴ – 2x³ + 4x² + x – 2

= 3x⁴ (x – 2) – 2x² ( x- 2) + 1 ( x – 2)

= (x – 2) (3x⁴ – 2x² + 1)

Question no – (8) 

Solution : 

Given, -x²y – x + 3xy + 3

= 3 – x + 3xy – x²y

= 1 (3 – x) + xy (3 – x)

= (3 – x) (1 + xy)

Question no – (9) 

Solution : 

Given, 6a² – 3a²b – bc² + 2c²

= 6a² – 3a²b + 2c² – bc²

= 3a² (2 – b) + c² (2 – b)

= (2 – b) (3a² + c²)

Question no – (10) 

Solution : 

As per the question, 3a²b – 12a² – 9b + 36

= 3a² (b – 4) – 9 (b – 4)

= (b – 4) (3a² – 9)

Question no – (11) 

Solution : 

Given, x² – (a – 3)x – 3a

= x² – ax + 3x – 3a

= x (x – a) + 3 (x – a)

= (x – a) (x + 3)

Question no – (12) 

Solution :

Given, x² – (b – 2)x – 2b

= x² – bx + 2x – 2b

= x (x – b) + 2 (x – b)

= (x – b) (x + 2)

Question no – (13) 

Solution : 

Given, a (b – c) – d(c – b)

= a(b – c) + d(b – c)

= (b – c) (a + d)

Question no – (14) 

Solution :

Given, ab² – (a – c) b – c

= ab² – ab + bc – c

= ab (b – 1) + c (b – 1)

= (b – 1) (ab + c)

Question no – (15) 

Solution :

Given, (a² – b²) c + (b² – c²) a

= a²c – b²c + ab² – ac²

= a²c – ac² + ab² – b²c

= ac (a- c) + b² (a – c)

= (a – c) (ac + b²)

Question no – (16) 

Solution :

Given, a³ – a² – ab + a + b – 1

= a² (a – 1) – b (a – 1) + 1 (a – 1)

= (a – 1) (a² – b + 1)

Question no – (17) 

Solution : 

Given, ab (c² + d²) – a²cd – b²cd

= abc² + abd² – a²cd – b²cd

= abc² – a²cd – b²cd + abd²

= ac (bc – ad) – bd (bc – ad)

= (bc – ad) (ac – bd)

Question no – (18) 

Solution : 

Given, 2ab² – aby + 2cby – cy²

= 2b (ab + cy) – y (ab + cy)

= (ab + cy) (2b – y)

Question no – (19) 

Solution : 

Given, ax + 2bx +3cx – 3a – 6b – 9c

= x (a + 2b + 3c) – 3 (a + 2b + 3c)

= (a + 2b + 3c) (x – 3)

Question no – (20) 

Solution : 

Given, 2ab²c – 2a + 3b³c – 3b – 4b²c² + 4c

= 2a(b²c – 1) + 3b (b²c – 1) – 4c (b²c – 1)

= (b²c – 1) (2a + 3b – 4c)

Factorisation Exercise 13(C) Solution :

Question no – (1) 

Solution : 

Given, 16 – 9x²

= (4)² – (3x)²

= (4 + 3x) (4 – 3x)

Question no – (2) 

Solution :

Given, 1 – 100a²

= (1)² – (10a)²

= (1 + 10a) ( 1 – 10a)

Question no – (3) 

Solution :

Given, 4x² – 81y²

= (2x)² – (9y)²

= (2x + 9y) (2x – 9y)

Question no – (4) 

Solution :

Given, 4/25 – 25b²

= (2/5)² – (5b)²

= (2/5 + 5b) (2/5 – 5b)

Question no – (5)  

Solution : 

Given, (a + 2b)² – a²

= (a + 2b + a) (a + 2b – a)

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

= 2 (a + b) (2b)

= 4b (a + b)

Question no – (6) 

Solution : 

Given, (5a – 3b)² – 16b²

= (5a – 3b)² – (4b)²

= (5a – 3b + 4b) (5a – 3b – 4b)

= (5a + b) (5a – 7b)

Question no – (7) 

Solution : 

Given, a⁴ – (a² – 3b²)²

= (a²)² – (a² – 3b²)²

= (a² + a² – 3b²) (a² – a² + 3b²)

= (2a² – 3b²) (3b²)

Question no – (8)

Solution : 

Given, (5a – 2b)² – (2a – b)²

= (5a – 2b + 2a – b) (5a – 2b – 2a + b)

= (7a – 3b) (3a – b)

Question no – (9)

Solution :

Given, 1 – 25 (a + b)²

= (1)² – [5 (a + b)]²

= [1 + 5 (a + b)] [1 – 5 (a + b)]

= (1 + 5a + 5b) (1 – 5a – 5b)

Question no – (10)

Solution : 

Given, 4(2a + b)² – (a – b)²

= [2 (2a + b)]² – (a – b)²

= [ 2(2a + b) + a – b] [ 2 (2a + b) – a + b]

= (4a + 2b + a – b) (4a + 2b – a + b)

= (5a + b) (3a + 3b)

= (5a + b) 3 (a + b)

= 3 (5a + b) (a + b)

Question no – (12) 

Solution : 

Given, 49(x – y)² – 9 (2x + y)²

= [7 (x – y)]² – [ 3 (2x + y)]²

= (7x – 7y)² – (6x + 3y)²

= (7x – 7y + 6x + 3y) (7x – 7y – 6x – 3y)

= (13x – 4y) (x – 10y)

Question no – (13) 

Solution :

Given, (6 2/3)² – (2 1/3)²

= (20/3)² – (7/3)²

= (20/3 + 7/3) (20/3 – 7/3)

= (20 + 7/3) (20 – 7/3)

= 27/3 × 13/3

= 39

Question no – (14) 

Solution :

Given, (7 3/10)² – (2 1/10)²

= (73/10)² – (21/10)²

= (73/10 + 21/10) (73/10 – 21/10)

= (73 + 21/40) (73 – 21/10)

= 94/10 × 52/10

= 4888/100

= 48 22/25

Question no – (15) 

Solution :

Given, (0.7)² – (0.3)²

= (0.7 + 0.3) (0.7 – 0.3)

= 1.0 × 0.4

= 0.4

Question no – (16) 

Solution : 

Given, (4.5)² – (1.5)²

= (4.5 + 1.5) (4.5 – 1.5)

= 6 × 3

= 18

Question no – (17) 

Solution : 

As per the question, 75 (x + y)² – 48 (x – y)²

= 75x² – 48b²   …..[x + y = a; x – y = b]

= 3 (25a² – 16b²)

= 3{(5a)² – (4b)²}

= 3 {(5a + 4b) (5a – 4b)²}

= 3 [{5 (x + y) + 4 (x – y)} {5 (x + y) – 4 (x – y)}]

= 3 [5x + 5y + 4x – 4y] [5x + 5y – 4x + 4y]

= 3(9x + y) (x + 9y)

Question no – (18) 

Solution : 

Given, a² + 4a + 4 – b²

= (a)² + 2 × a × 2 + (2)² – (b)²

= (a + 2)² – (b)²

= (a + 2 + b) (a + 2 – b)

Question no – (19) 

Solution : 

Given, a² – b² – 2b – 1

= a² – (b² + 2b + 1)

= a² – (b + 1)²

= (a + b + 1) (a – b – 1)

Question no – (20) 

Solution : 

Given, x² + 6x + 9 – 4y²

= x² + 2x . 3 + 3² – (2y)²

= (x + 3)² – (2y)²

= (x + 3 + 2y) (x + 3 – 2y)

Factorisation Exercise 13(D) Solution :

Question no – (1) 

Solution : 

Given, x² + 6x + 8

= x² + 4x + 2x + 8

= x (x + 4) + 2 (x + 4)

= (x + 4) (x + 2)

Question no – (2) 

Solution : 

Given, x² + 4x + 3

= x² + 3x + x + 3

= x (x + 3) + 1(x + 3)

Question no – (3) 

Solution :  

Given, a² + 5a + 6

= a² + 3a + 2a + 6

= a(a + 3) + 2(a + 3)

= (a + 3) (a + 2)

Question no – (4) 

Solution : 

Given, a² – 5a + 6

= a² – 3a – 2a + 6

= a (a – 3) – 2 (a – 3)

= (a – 3) (a – 2)

Question no – (6) 

Solution :

Given, x² + 5xy + 4y²

= x² + 4xy + xy + 4y²

= x (x + 4y) + y (x + 4y)

= (x + y) (x + 4y)

Question no – (7) 

Solution : 

Given, a² – 3a – 40

= a² – 8a + 5a – 40

=  a (a – 8) + 5(a – 8)

= (a – 8) (a + 5)

Question no – (8) 

Solution : 

Given, x² – x – 72

= x² – 9x + 8x – 72

= x (x – 9) + 8 (x – 9)

= (x – 9) (x + 8)

Question no – (9) 

Solution : 

Given, x² – 10xy + 24y²

= x² – 6xy – 4xy + 24y²

= x (x – 6y) – 4y (x – 6y)

= (x – 6y) (x – 4y)

Question no – (10) 

Solution : 

Given, 2a² + 7a + 6

= 2a² + 4a + 3a + 6

= 2a (a + 2) + 3 (a + 2)

= (2a + 3) (a + 2)

Question no – (11) 

Solution : 

Given, 3a² – 5a + 2

= 3a² – 3a – 2a + 2

= 3a (a – 1) -2 (a – 1)

= (a – 1) (3a – 2)

Question no – (12) 

Solution : 

Given, 7b² – 8b + 1

= 7b² – 7b – b + 1

= 7b(b – 1) – 1(b – 1)

= (b – 1) (7b – 1)

Question no – (13) 

Solution : 

Given, 2a² – 17ab + 26b²

= 2a² – 13ab – 4ab + 26b²

= a (2a – 13b) – 2b (2a -13b)

= (a – 2b) (2a – 13b)

Question no – (14) 

Solution : 

Given, 2x² + xy – 6y²

= 2x² + 4xy – 3xy – 6y²

= 2x (x + 2y) – 3y (x + 2y)

= (2x – 3y) (x + 2y)

Question no – (15) 

Solution : 

Given, 4c² + 3c – 10

= 4c² + 8c – 5c – 10

= 4c (c + 2) – 5 (c + 2)

= (4c – 5) (c + 2)

Question no – (16) 

Solution : 

Given, 14x² + x – 3

= 14x² + 7x – 6x – 3

= 7x (2x + 1) – 3 (2x + 1)

= (2x + 1) (7x – 3)

Question no – (17) 

Solution : 

Given, 6 + 7b – 3b²

= 6 + 9b – 2b – 3b²

= 3 (2 + 3b) – b (2 + 3b)

= (2 + 3b) (3 – b)

Question no – (18)

Solution : 

Given, 5 + 7x – 6x²

= 5 + 10x – 3x – 6x²

= 5 (1+ 2x) – 3 (1 + 2x)

= (1 + 2x) (5 – 3)

Question no – (19) 

Solution :

Given, 4 + y – 14y²

= 4 + 8y – 7y – 14y²

= 4 (1 + 2y) – 7y (1 + 2y)

= (1 + 2y) (4 – 7y)

Question no – (20) 

Solution :

Given, 5 + 3a – 14a²

= 5 + 10a – 7a – 14a²

= 5 (1 + 2a) – 7a (1 + 2a)

= (5 – 7a) (1 + 2a)

Question no – (21) 

Solution :

Let, 2a + b = x

= x² + 5x + 6

= x² + 3x + 2x + 6

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

= (x + 3) (x + 2)

= (2a + b + 3) (2a + b + 2)

Question no – (23) 

Solution :

Given, (x – 2y)² – 12 (x – 2y) + 32

Let, x – 2y = a

= a² – 12a + 32

= a² – 8a – 4a + 32

= a(a – 8) – 4 (a – 8)

= (a – 8) ( a – 4)

= (x – 2y – 8) (x – 2y – 4)

Question no – (24) 

Solution : 

Given, 8 + 6 (a + b) – 5 (a + b)²

Let, a + b = x

= 8 + 6x – 5x²

= 5x² – 6x – 8

= 5x² – 10x + 4x – 8

= 5x (x – 2) + 4 (x – 2)

= (x – 2) (5x + 4)

= (a + b – 2) (5 (a + b) + 4)

= (a + b – 2) (5a + 5b + 4)

= (2 – a – b) (4 + 5a + 5b)

Question no – (25) 

Solution : 

Given, 2(x + 2y)² – 5 (x + 2y) + 2

Let, x + 2y = a

= 2a² – 5a + 2

= 2a² – 4a – a + 2

= 2a (a – 2) – 1 (a – 2)

= (a – 2) (2a – 1)

= (x + 2y – 2) (2 (x + 2y) -1)

= (x + 2y – 2) (2x + 4y – 2)

Factorisation Exercise 13(E) Solution :

Question no – (1) 

Solution :

(i) Given, x² + 14x + 49

= x² + 2.x.7 + 7²

= (x + 7)²

∴ It is a perfect square.

(ii) a² – 10a + 25

= a² – 2.a.5 + 5²

= (a – 5)²

∴ It is a perfect square.

(iii) 4x² + 4x + 1

= (2x)² + 2. 2x. 1 + 1

= (2x + 1)²

∴ It is a perfect square.

(iv) 9b² + 12b + 16

= (3b)² + 2.3b.2b + 4²

∴ It is not perfect square.

(v) 16x² – 16xy + y²

= (4x)² – 4.4x + y + y²

It can’t express of square.

So, it is not a perfect square.

(vi) x² – 4x + 16

= x² – 2.x.2 + 4

It can’t express of square

Thus, it is not a perfect square.

Question no – (2)

Solution : 

Given, 2 – 8x²

= 2 [1 – 4x²]

= 2 [(1)² – (2x)²]

= 2 (1 + 2x) (1 – 2x)

Question no – (3) 

Solution : 

Given, 8x²y – 18y³

= 2y (4x² – 9y)

= 2y {(2x)² – (3y)²}

= 2y (2x + 3y) (2x – 3y)

Question no – (4) 

Solution : 

ax² – ay²

= a (x² – y²)

= a (x + y) (x – y)

Question no – (5) 

Solution : 

Given, 25x³ – x

= x (25x² – 1)

= x {(5x)² – (1)²}

= x (5x + 1) (5x – 1)

Question no – (6) 

Solution : 

Given, a⁴ – b⁴

= (a²)² – (b²)²

= (a² + b²) (a² – b²)

Question no – (7) 

Solution : 

Given, 16x⁴ – 81y⁴

= (4x)² – (9y)²

= (4x² + 9y²) (4x² – 9y²)

= (4x² + 9y²) {(2x) ² – (3y) ²}

= (4x² + 9y²) (2x + 3y) (2x – 3y)

Question no – (8) 

Solution : 

Given, 625 – x⁴

= (25)² – (x)²

= (25 + x²) (25 – x²)

= (25 + x²) {(5)² – (x)²}

= (25 + x²) (5 + x) (5 – x)

Question no – (9) 

Solution : 

Given, x² – y² – 3x – 3y

= (x + y) (x – y) – 3 (x + y)

= (x + y) (x – y – 3)

Question no – (10) 

Solution : 

Given, x² – y² – 2x + 2y

= (x + y) (x – y) – 2 (x – y)

= (x – y) (x + y – 2)

Question no – (11) 

Solution : 

Given 3x² + 15x – 72

= 3 (x² + 5x – 24)

= 3 [x² + 8x – 3x – 24]

= 3 [x (x + 8) – 3 (x + 8)]

= 3 [(x + 8) (x – 8)]

Question no – (12) 

Solution : 

Given, 2a² – 8a – 64

= 2 (a² – 4a – 32)

= 2 (a² – 8a + 4a – 32)

= 2 {a (a – 8) + 4 (a – 8)}

= 2 (a – 8) (a + 4)

Question no – (13) 

Solution : 

Given, 5b² + 45b + 90

= 5 (b² + 9b + 18)

= 5 (b² + 6b + 3b + 18)

= 5 {b (b + 6) + 3 (b + 6)}

= 5 (b + 6) (b + 3)

Question no – (14)  

Solution : 

Given, 3x²y + 11xy + 6y

= y (3x² + 11x + 6)

= y [3x² + 9x + 2x + 6]

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

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

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

Question no – (15)  

Solution : 

Given, 5ap² + 11ap + 2a

= a [5p² + 11p + 2]

= a [5p² + 10p + p + 2]

= a [5p (p + 2) + 1 (p + 2)]

= a (p + 2) (5p + 1)

Question no – (16) 

Solution : 

Given, a² + 2ab + b² – c²

= (a + b)² – c²

= (a + b + c) (a + b – c)

Question no – (17) 

Solution : 

Given, x² + 6xy + 9y² + x + 3y

= (x + 3y)² + (x + 3y)

= (x + 3y) (x + 3y + 1)

Question no – (18) 

Solution : 

Given, 4a² – 12ab + 9b² + 4a – 6b

= [(2a)² – 2.2a.3b + (3b)²] + (2a – 3b)

= (2a – 3b)² + 2 (2a – 3b)

= (2a – 3b) (2a – 3b + 2)

Question no – (19) 

Solution : 

Given, 2a²b² – 98b⁴

= 2b² (a² – 49b²)

= 2b² {(a)² – (7b)²}

= 2b²(a + 7b) (a – 7b)

Question no – (20) 

Solution : 

Given, a² – 16b² – 2a – 8b

= a² – (4b)² – 2 (a – 4b)

= (a + 4b) (a – 4b) – 2 (a – 4b)

= (a – 4b) (a + 4b – 2)

Factorisation Exercise 13(F) Solution :

Question no – (1) 

Solution : 

(i) 6x³ – 8x²

= 2x²(3x – 4)

(ii) 35a³b²c + 42ab²c²

= 7ab²c (5a² + 6c)

(iii) 36x²y² – 30x³y³ + 48x³y²

= 6x²y² (6 – 5xy + 8xy)

(iv) 8 (2a + 3b)³ – 12 (2a + 3b)²

= 4 (2a + 3b)² [ 2(2a + 3b) – 3]

= 4 (2a + 3b)² [4a + 6b – 3]

(v) 9a (x – 2y)⁴ – 12a (x – 2y)³

= 3a (x – 2y)³ [3 (x – 2y) – 4]

= 3a (x – 2y)³ (3a – 6y – 4)

Question no – (2) 

Solution : 

(i) a² – ab – 3a + 3b

= a (a – b) – 3 (a – b)

= (a – b) (a – 3)

(ii) xy² – xy² + 5x – 5y

= xy (x – y) + 5 (x – y)

= (x – y) (xy + 5)

(iii) a² – ab (1 – b) – b³

= a² – ab + ab² – b³

= a (a – b) + b² (a – b)

= (a – b) (a + b²)

(iv) xy² + (x – 1) y – 1

= xy² + xy – y – 1

= xy (y + 1) – 1 (y + 1)

= (xy – 1) (y + 1)

(v) (ax + by)² + (bx – ay)

= a²x² + b²y² + 2abxy + b²x² + ay – 2abxy

= a²x² + b²y² + b²x² + a²y²

= a² (x + y²) + b² (y² + x²)

(vi) ab(x² + y²) – xy(a² + b²)

= abx² + aby² – a²xy – b²xy

= abx² – a²xy – b²xy + aby²

= ax (bx – ay) – by (bx – ay)

= (bx – ay) (ax – by)

(vii) m – 1 – (m – 1)² + am – a

= (m – 1) – (m – 1)² + a (m – 1)

= (m – 1) (1 – m + 1 + a)

= (m – 1) (2 – m + a)

Question no – (3) 

Solution : 

(i) a² – (b – c)²

= (a + b – c) (a – b + c)

(ii) 25(2x – y)² – 16 (x – 2y)

= {5 (2x – y)²} – (4 (x – 2y)²

= [5 (2x – y) – 4 (x – 2y)] [5 (2x – y) + 4 (x – 2y)]

= (10x – 5y – 4x + 8y) (10x – 5y + 4x – 8y)

= (6x + 3y) (14x – 13y)

= 3 (2x + y) (14x – 13y)

(iv) 9x² – 1/16

= (3x)² – (1/4)²

= (3x + 1/4) (3x – 1/4)

(v) 25 (x – 2y)² – 4

= (5 (x – 2y)² – 2²

= [5 (x – 2y) – 2] [5 (x – 2y) + 2]

= [5 x – 10y – 2] [5x – 10y + 2]

Question no – (4) 

Solution : 

(i) a² – 23a + 42

= a² – 21a – 2a + 42

= a (a – 21) – 2 (a – 21)

= (a – 2) (a – 21)

(ii) a² – 23a – 108

= a² – 27a + 4a – 108

= a (a – 27) + 4 (a – 27)

= (a – 27) (a + 4)

(iii) 1 – 18x – 63x²

= 1 – 21x + 3x – 63x²

= 1 (1 – 21x) + 3x(1 – 21x)

= (1 + 3x) (1 – 21x)

(iv) 5x² – 4xy – 12y²

= 5x² – 10xy + 6xy – 12y²

= 5x (x – 2y) + 6y (x – 2y)

= (x – 2y) (5x + 6y)

(v) x(3x + 14) + 8

= 3x² + 14x + 8

= 3x² + 12x + 2x + 8

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

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

(vi) 5 – 4x (1 + 3x)

= 5 – 4x – 12x²

= 5 – 10x + 6x – 12x²

= 5 – 10x + 6x (1 – 2x)

= (5 + 6x) (1 – 2x)]

(vii) x²y² – 3xy – 40

= x²y² – 8xy + 5xy – 40

= xy (xy – 8) + 5 (xy – 8)

= (xy – 8) (xy + 5)

(viii) (3x – 2y)² – 5(3x – 2y) – 24

Let, 3x – 2y = a

∴ a² – 5a – 24

= a² (a – 8) + 3(a – 8)

= (a + 3) (a – 8)

= (3x – 4y + 3) (3x – 4y – 8)

(ix) 12 (a + b)² – (a + b) – 35

Let, a + b = x

∴ 12x² – x – 35

= 12x² – 21x + 20x – 35

= 3x (4x – 7) + (4x – 7)

= (4x – 7) (3x + 5)

= {4 (a + b) – 7} {3 (a + b) + 5}

= (4a + 4b – 7) (3a + 3b + 5)

Question no – (6)

Solution : 

(i) 2a³ – 50a

= 2a (a² – 25)

= 2a (a + 5) (a – 5)

(ii) 54a²b² – 6

= 6 (9a²b² – 1)

= 6 {(3ab)² – 1²}

= 6 (3ab + 1) (3ab – 1)

(iii) 64a2b – 144b³

= 16b (4a² – 9b²)

= 16b{(2a)² – (3b)²}

= 16b (2a + 3b) (2a – 3b)

(iv) (2x – y)³ – (2x – y)

= (2x – ) [ (2x – y)^{2 – 1}

= (2x – y) (2x – y + 1) (2x – y – 1)

(v) x² – 2yz + y² – z²

= x² – (y² + 2yz + z)

= x² – (y + z)²

= (x + y + z) (x – y – z)

(vi) 7a⁵ – 567a

= 7a (a⁴ – 81)

= 7a {(a²)² – 92}

= 7a (a² + 9) (a² – 9)

= 7a (a² + 9) (a² + 3) (a – 3)

(vii) 5x² – 20×4/9

= 5x² [1 – 4×2/9]

= 5x² [1 – 2x/3] [1 + 2x/3]

Question no – (7) 

Solution : 

As per the given question,

xy² – xz²

= x (x² – z²)

= x (y + z) (y – z)

Now, (i) 9 × 8² – 9 × 2²

= 9 (8² – 2²)

= 9(8 + 2) (8 – 2)

= 9 × 10 × 6

= 540

(ii) 40 × 5.5² – 40 × 4.5²

= 40 (5.5² – 4.5²)

= 40 (5.5 + 4.5) (5.5 – 4.5)

= 40 × 10 × 1

= 400

Question no – (8)

Solution : 

(i) (a – 3b)² – 36b²

= (a – 3b)² – (6b)²

= (a – 3b + 6b) (a – 3b – 6b)

= (a + 3b) (a – 9b)

(ii) 25 (a – 5b)² – 4 (a – 3b)²

= [5 (a – 5b)] – [2 (a – 3b)]²

= (5a – 2b)² – (2a – 6b)²

= (5a – 25b + 2a – 6b) (5a – 25b – 2a + 6b)

= (7a – 31b) (3a – 19b)

(iii) a² – 0.36 b²

= a² – (0.6b)²

= (a + 0.6b) (a – 0.6b)

(iv) a⁴ – 625

= (a²)² – (25)²

= (a² + 25) (a² – 25)

= (a² + 25) {(a)² – (5)²}

= (a² + 25) (a + 5) (a – 5)

(v) x⁴ – 5x² – 36

= (x²)² – 5x² – 36

= (x²)² – 9x² + 4x² – 36

= x2 (x² – 9) + 4 (x² – 9)

= (x² + 4) (x² – 9)

= (x² + 4) (x + 3) (x – 3)

Question no – (9) 

Solution : 

a²b – b³

= b (a2 – b2)

= b (a + b) (a – b)

∴ 100² × 100 – 100³

= 100 (101² – 100²)

= 100 (101 + 100) (101 – 100)

= 100 (201) × 1

= 20100

Therefore, the value will be 20100.

Question no – (10) 

Solution : 

301² × 300 – 300³ …according to the question,

= 300 (301² – 300²)

= 300 (301 + 300) (301 – 300)

=  300 (601) × 1

= 180300

Next Chapter Solution : 

👉 Chapter 14 👈