# OP Malhotra Class 9 ICSE Maths Solutions Chapter 4

## OP Malhotra Class 9 ICSE Maths Solutions Chapter 4 Factorisation

Welcome to NCTB Solutions. Here with this post we are going to help 9th class students for the Solutions of OP Malhotra Class 9 ICSE Math Book, Chapter 4, Factorisation. Here students can easily find step by step solutions of all the problems for Factorisation, Exercise 4a, 4b, 4c, 4d, 4e, 4f, 4g and 4h 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 4 solutions. Here in this post all the solutions are based on ICSE latest Syllabus.

Factorisation Exercise 4(a) Solution :

Question no – (1)

Solution :

Given, 8x + 16y

= 8 (x + 2y)

Question no – (2)

Solution :

Given, ax – 2ay

= a ( x – 2y)

Question no – (3)

Solution :

Given, 25xy – 50y

= y (25x – 50)

= 25y (x – 2)

Question no – (4)

Solution :

Given, abc – ac

= abc – ac

= ac (b – 1)

Question no – (5)

Solution :

Given, – 121m – 11 m2

= – 121 – 11m2

= 11 (m – 11 – m)

Question no – (6)

Solution :

Given, a2b + ab2

= ab = (a + b)

Question no – (7)

Solution :

Given, 24x3y – 30x2y

= 6xy2 (4x – 5)

= 6x2y (4x – 5)

Question no – (8)

Solution :

Given, – 9x3 – 12x2 + 3x

= (- 3x2 – 4x + 1)

Question no – (9)

Solution :

Given, 25abc3 – 35ab2c3 + 45a3bc2

= 25abc3 – 35ab2c3 + 45a3bc2

= 5abc2 (5ac – 7bc + 9a2)

Question no – (11)

Solution :

Given, x (x – y) + (x – y)

= x (x – y) + (x – y)

= (x – y) b(x + 1)

Question no – (12)

Solution :

Given, (x + y) – (x + y)2

= (x + y) – (x + y)2

= (x + y) (1 – x – y)

Question no – (13)

Solution :

Given, a (x – y) – b (y – x)

= a (x – y) – b (y – x)

= a (x – y) + b (x – y)

= (x – y) (a + b)

Question no – (14)

Solution :

Given, (a – b)4 + (a – b)3

= (a – b)4 + (a – b)3

= (a – b)3 + (a – b + 1)

Question no – (15)

Solution :

Given, (x – y)6 + (y – x)5

= (x – y)6 – (x – y)5

= (x – y)5 (x – y – 1)

Factorisation Exercise 4(b) Solution :

Question no – (1)

Solution :

Given, bx + 2b + cx + 2c

= b (x + 2) + c (x + 2)

= (x + 2) + c (x + 2)

Question no – (2)

Solution :

Given, y3 + 2y2 + 3y + 6

= y2 (y + 2) + 3y + 6

= (y2 + 3) (y + 2)

Question no – (3)

Solution :

Given, xa + 3b + xb + 3a

= xa + xb + 3b + 3a

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

= (a + b) (x + 3)

Question no – (4)

Solution :

Given, 8xy + 5zy – 8xt – 5zt

= y (8y + 52) – t (8x + 5x)

= (y – t) (8x + 5z)

Question no – (5)

Solution :

Given, 8kl + 12ml – 12mn – 8km

= 8kl – 8km + 12ml – 12m/n

= 8k (l – n) + 121m (l – n)

= (l – n) (8k + 12m)

Question no – (6)

Solution :

Given, 32 (x + y)2 – 2x – 2y

= 32 (x + y)2 – 2 (x + y)

Let, x + y = a

= 32a2 – 2a

= 2a (16a – 1)

= 2 (x + y) (16x + 16y – 1)

Question no – (7)

Solution :

Given, x3 – x2 + ax + x – a – 1

= x2 (x – 1) + a (x – 1) + 1 (x – 1)

= (x – 1) (x2 + a + 1)

Question no – (8)

Solution :

Given, ab (c2 + 1) + c(a2 + b2)

= abc2 + ab + a2c + b2c

= abc2 + a2c + b2c + ab

= ac (bc + a) + b (bc + a)

= (ac + b) (bc + a)

Question no – (9)

Solution :

Given, a3 – a2 + xa + a – x – 1

= a3 – a2 + xa – x + a – 1

= a2 (a – 1) + x (a – 1) + 1 (a – 1)

= (a – 1) (a2 + x + 1)

Question no – (10)

Solution :

Given, 6a3b + 3a2b2 – 2a2b – ab2

= 3a2b (2a + b) – ab (2a + b)

= (2a + b) (3a2b – ab)

= (2a + b) ab (3a – 1)

Question no – (11)

Solution :

Given, a3 + ab (1 – 2a) – 2b2

= a3 + ab – 2a2b – 2b2

= a (a+ b) – 2b (a2 + b)

= (a2 + b) (a – 2b)

Question no – (12)

Solution :

Given, (p2 + 1)q – p2 – q2

= p2q + q – p2 – q2

= p2q – p2 – q2 + q

= p2 (q – 1) – q (q – 1)

= (q – 1) (p2 – q)

Factorisation Exercise 4(c) Solution :

Question no – (1)

Solution :

Given, x2 + 4x + 4

= x2 + 2.x.2 + 22

= (x + 2) 2

Question no – (2)

Solution :

Given, x2 + 6x + 9

= x2 + 2.x.3 + 32

= (x + 3)2

Question no – (3)

Solution :

Given, x2 – 10x + 25

= x2 – 2.x.5 + 52

= (x – 5)2

Question no – (4)

Solation :

Given, 4x2 – 4x + 1

= -(2x)2 – 2.x.1 + 12

= (2x – 1)2

Question no – (5)

Solation :

Given, 1 – 8x + 16x2

= (1)2 – 2.1.4x + (4x)2

= (1 – 4x)2

Question no – (6)

Solation :

Given, 49x4 + 168x2y2 + 144y4

= (7x2)2 + 2.7x.12y2 + (12y2)2

= (7x2 + 12y2)2

Question no – (7)

Solation :

Given, x2 + x + 1/4

= x2 + 2.x. 1/2 + (1/2)2

= (x + 1/2)2

Question no – (8)

Solution :

Given, 25p2 + 5p/2q + 1/16q2

= (5p)2 + 2.5p. 1/4q + (1/4q)2

= (5p + 1/4q)2

Factorisation Exercise 4(d) Solution :

Question no – (1)

Solution :

Given, x2 – 4

= (x)2 – (2)2

= (x + 2) (x – 2)

Question no – (2)

Solution :

Given, y2 – 25

= (y)2 (5)2

= (y + 5) (y – 5)

Question no – (3)

Solution :

Given, a2 – 1

= (a)2 – (1)2

= (a + 1) (a – 1)

Question no – (4)

Solution :

Given, 9x2 – 64

= (3z)2 – (2)2

= (3x + 8) (3z – 8)

Question no – (5)

Solution :

Given, 9x2 – b2

= (3x)2 – b2

= (3x + b) (3x – b)

Question no – (6)

Solution :

Given, 25 – x2y2

= (5)2 – (xy)2

= (5 + xy) (5 – xy)

Question no – (7)

Solution :

Given, 81a2x2 – 49b2y2

= (99x)2 – (7by)2

= (9ax + 7by) (9ax – 7by)

Question no – (8)

Solution :

Given, x2 – 1/4

= (x)2 – (1/2)2

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

Question no – (9)

Solution :

Given, -25 + 1/64b2

= (1/64b2 – 25)

= (1/8b)2 – (5)2

= (1/8b + 5) (1/8b – 5)

Question no – (10)

Solution :

Given, a2/9 – b2/16

= (a/3)(b/4)2

= (a/3 + b/4) (a/3 – b/4)

Question no – (11)

Solution :

Given, 2.25a2 – b2

= (1.5a)2 – b2

= (1.5a + b) (1.5 – b)

Question no – (12)

Solution :

Given, 36a8 – 121

= (6a4)2 – (11)2

= (6a4 + 11) (6a4 – 11)

Question no – (13)

Solution :

Given, 542 – 362

= (54 + 36) (54 – 36)

= 90 × 18

= 1620

Question no – (14)

Solution :

Given, (3.2)2 – (1.8)2

= (3.2 + 1.8) (3.2 – 1.8)

= 5.0 × 1.4

= 7

Factorisation Exercise 4(e) Solution :

Question no – (1)

Solution :

Given, 7x2 – 7

= 7 (x2 – 1)

= 7 (x + 1) (x – 1)

Question no – (2)

Solution :

Given, 8 – 50y2z2

= 2 (4 – 25y2x2)

= {(2)2 – (5y2z2)

= 2{(2 + 5yz) (2 – 5yz)

= 2 (2 – 5yz) (2 + 5yz)

Question no – (3)

Solution :

Given, ab2 – ac2

= a (b2 – c2)

= a (b + c) (b – c)

Question no – (4)

Solution :

Given, 36x3 – x

= x (36x2 – 1)

= x {(6x)2 – 12

= x (6x + 1) (6x – 1)

Question no – (5)

Solution :

Given,  x(x2 – 1) + 7 (x2 – 1)

= x3 – x + 7x2 – 7

= (x2 – 1) (x + 7)

= (x + 1) (x – 1) (x + 7)

Question no – (7)

Solution :

Given, 5c2 (c + 2)2 – 45(c + 2)2

= 5 (c + 2)2 [c2 – 9)

= 5 (c + 2)2 {(c)2 – (3)2}

= 5 (c + 2)2 (c + 3) (c – 3)

Question no – (8)

Solution :

Given, (a + b)2 – 1

= (a + b)2 – (1)2

= (a + b + 1) (a + b – 1)

Question no – (10)

Solution :

Given, 25a2 – 17(x – y)2

= (5a)2 – {4 (x – y)}2

= [5a + 4 (x – y) [5a – 4 (x – y)]

Question no – (11)

Solution :

Given in the question, 20 – 45 (m + n)2

∴ 20 – 45 (m + n)2

= 5 [4 – 9 (m + n)2

= 5[(2)– {3 (m + n)}2

= 5 [2 + 3 (m + n)] [2 – 3 (m + n)]

Question no – (12)

Solution :

Given, x4 – y4

= (x2)2 – (y2)2

= (x2 + y2) (x2 – y2)

= (x2 + y2) (x + y)( (x – y)

Question no – (13)

Solution :

Given, x4 – 625

= (x2)2 – (25)2

= (x2 + 25) (x2 – 25)

= (x2 + 25) (x2 – 25)

= (x2 + 25) {(x)2 – (5)2

= (x2 + 25) (x + 5) (x – 5)

Question no – (14)

Solution :

In the given question, xy5 – yx5

= xy [y4 – x4]

= xy [(y2 + x2) (y2 – x2)

= xy [(y2 + x2) (y + x) (y – x)

Question no – (15)

Solution :

Given, 81x4 – 256y4

= (9x2) – (16y2)2

= (9x2 + 16y2) (9x2 – 16y2)

= (9x2 + 16y2) {(3x)2 – 94y)2}

= (9x2 + 16y2) (3x + 4y) (3x – 4y)

Question no – (16)

Solution :

Given, a2 + ac + bc – b2

= a2 – b2 + ac + bc

= (a + b) (a – b) + c (a + b)

= (a + b) (a – b + c)

Question no – (17)

Solution :

Given, 4a2 – b2 + 2a + b

= (2a)2 – (b)2 + 2a + b

= (2a + b) (2a – b) (2a + b)

= (2a + b) (2a – b + 1)

Question no – (18)

Solution :

In the given question, x2 + 3x – y2 – 3y

∴ x2 + 3x – y2 – 3y

= x2 – y2 + 3x – 3y

= (x + y) (x – y) + 3 (x – y)

= (x + y + 3) (x – y)

Question no – (19)

Solution :

Given in the question, a2 + b2 – 2ab – 4c2

∴ a2 + b2 – 2ab – 4c2

= (a – b)2 – 4c2

= (a – b)2 – (2c)2

= (a – b + 2c) (a – b – 2c)

Question no – (20)

Solution :

In the question we get, 9x2 – 6xy + y2 – x2

∴ 9x2 – 6xy + y2 – x2

= [(3x)2 – 2.3x.y + y2] – z2

= (3x – y)2 – z2

= (3x – y + z) (3x – y – z)

Question no – (21)

Solution :

As per the question, x2 – 1 – 2a – a2

∴ x2 – 1 – 2a – a2

= x2 – (1 + 2a + a2)

= x2 – (1 + a)2

= (x + 1 + a) (x – 1 – a)

Question no – (22)

Solution :

Given, 4a2 + b2 – c2 + 4ab

∴ 4a2 + b2 – c2 + 4ab

= 4a2 + b2 + 4ab – c2

= (2a)2 + (b)2 + 2.2a.b – c2

= (2a + b)2 – (c)2

= (2a + b + c) (2a + b – c)

Question no – (23)

Solution :

Given in the question, x3 + 2x2 – x – 2

∴ x3 + 2x2 – x – 2

= x2 (x + 2) – 1 (x + 2)

= (x2 – 1) (x + 2)

= (x + 1) (x – 1) (x + 2)

Question no – (24)

Solution :

Given in the question, 1 + 2ab – (a2 + b2)

∴ 1 + 2ab – (a2 + b2)

= 1 + 2ab – a2 – b2

= 1 (a2 – 2ab + b2)

= 12 – (a – b)2

= (1 + a – b) (1 – a + b)

Question no – (25)

Solution :

Given, x2 + 1/x2 – 11

∴ x2 + 1/x2 – 11

= x2 + 1/x2 – 2.x 1/x – 11

= (x2 – 1/x2) – 9

= (x – 1/x)2 – (3)2

= (x – 1/x + 3) (x – 1/x – 3)

Question no – (26)

Solution :

As per the question, x4 + 3x2 + 4

∴ x4 + 3x2 + 4

= x4 + 4 + 3x2

= (x) + 22 – 2.x2.2 + 3x2

= (x2 + 2) – x2

= (x2 + 2 + x) (x2 + 2 – x)

Factorisation Exercise 4(f) Solution :

Question no – (1)

Solution :

In the given question, a2 + 5a + 6

∴ a2 + 5a + 6

= a2 + 3a + 2a + 6

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

= (a + 3) (a + 2)

Question no – (2)

Solution :

Given in the question, a2 + 6a + 8

∴ a2 + 6a + 8

= a2 + 4a + 2a + 8

= a (a + 4) + 2 (a + 4)

= (a + 4) (a + 2)

Question no – (3)

Solution :

From the question we get, p2 + 10p + 16

∴ p2 + 10p + 16

= p2 + 2b + 8p + 16

= p (p + 2) + 8 (p + 2)

= (p + 2) (p + 8)

Question no – (4)

Solution :

In the question we get, a2 + 13a + 42

∴ a2 + 13a + 42

= a2 + 7a + 6a + 42

= a (a + 7) + 6 (a + 7)

= a (a + 7) (a + 6)

Question no – (5)

Solution :

Given in the question, a2 + 25a – 54

∴ a2 + 25a – 54

= a2 + 27a – 2a – 54

= a(a + 27) – 2 (a + 27)

= (a + 27) (a – 2)

Question no – (6)

Solution :

In the question we get, x2 + 5x – 176

∴ x2 + 5x – 176

= x2 + 16x – 11x – 176

= x (x + 16) – 11 (x + 16)

= (x + 16) (x – 11)

Question no – (7)

Solution :

Given, y2 – 18y + 65

∴ y2 – 18y + 65

= y2 – 13y – 5y + 65

= y (y – 13) – 5 (y – 13)

= (y – 13) – 5 (y – 13)

= (y – 13) (y – 5)

Question no – (8)

Solution :

Given, m2 – 29m + 204

∴ m2 – 29m + 204

= m2 – 17 – 13m + 204

= m (m – 17) – 12 (m – 17)

= (m – 17) (m – 12)

Question no – (9)

Solution :

In the question we get, b2 – 2b – 48

∴ b2 – 2b – 48

= b2 – 8b + 6b – 48

= b (b – 8) + 6 (b – 8)

= (b – 8) (b + 6)

Question no – (10)

Solution :

In the question, x2 – 11x – 102

∴ x2 – 11x – 102

= x2 – 17x + 6x – 102

= x (x – 17) + 6 (x – 17)

= (x – 17) (x + 6)

Question no – (11)

Solution :

Given in the question, 3 – 4t + t2

3 – 4t + t2

= 3 – 3t – t + t2

= 3 (1 – t) – t (1 – t)

= (3 – tr) (1 – t)

Question no – (12)

Solution :

From the question we get, 51 – 20k + k2

51 – 20k + k2

= k2 – 20k + 51

= k2 – 17k – 3k + 51

= k (k – 17) – 3 (k – 17)

= (k – 17) (k – 3)

Question no – (13)

Solution :

In the question we get, 2x2 – 10x + 12

∴ 2x2 – 10x + 12

= 2 (x2 – 5x + 6)

= 2 (x2 – 3x – 2x + 6)

= 2{x (x – 3) – 2 (x – 3)

= 2 (x – 3) (x – 2)

Question no – (14)

Solution :

Given in the question, 3x3 – 33x3 + 84x

Now, = 3x3 – 33x3 + 84x

= 3x (x2 – 11x + 28)

= 3x [x2 – 7x – 4x + 28)

= 3x [x (x – 7) – 4 (x – 7)]

= 3x (x – 7) (x – 4)

Question no – (15)

Solution :

In the question, 6y2 – 45y – 110

∴ 6y2 – 45y – 110

= 5 (y2 – 9y – 22)

= 5 (y2 – 11y + 2y – 22)

= 5y (y – 11) + 2 (y – 11)

= 5 (y – 11) (y + 2)

Question no – (16)

Solution :

From the question, x4 – 13x2 + 36

∴ x4 – 13x2 + 36

= x4 – 9x2 – 4x2 + 36

= x2 (x2 – 9) – 4 (x2 – 9)

= (x2 – 9) (x2 – 4)

= (x + 3) (x – 3) (x + 2) )(x – 2)

Question no – (17)

Solution :

Given in the question,  x2 + 3xy – 88y2

∴ x2 + 3xy – 88y2

= x2 + 11xy – 8xy – 88y2

= x (x + 11y) – 8y (x + 11y)

= (x + 11y) (x – 8y)

Question no – (18)

Solution :

In the question we get, x4 – x2y2 – 72y4

∴ x4 – x2y2 – 72y

= x4 – 9x2y2 + 8xy2 – 72y4

= x2(x2 – 9y2) + 8y2 (x2 – 9y2)

= (x2 + 8y2) (x2 – 9y2)

= (x2 + 8y2) (x + 3y) (x – 3y)

Question no – (19)

Solution :

Given in the question, a3b3 – 9a2b2 + 20ab

∴ a3b3 – 9a2b2 + 20ab

= ab (a2b2 – 9ab + 20)

= ab [a2b2 – 4ab – 5ab + 20]

= ab [ab (ab – 4) – 5 (ab – 4)

= ab (ab – 4) (ab – 5)

Question no – (20)

Solution :

From the question we get, (x2 + x)2 + 4 (x2 + y) = 21

Let, x2 + x = a

= a2 + 4a – 21

= a2 – 7a 3a – 22

= a (a – 7) + 3 (a – y)

= (a – y) (a + 3)

Factorisation Exercise 4(g) Solution :

Question no – (1)

Solution :

Given in the question, 2x2 + 3x + 1

∴ 2x2 + 3x + 1

= 2x2 + x + 2x + 1

= 2x2 + 2x + x + 1

= 2x (x + 1) + 1 (x + 1)

= (x + 1) (x + 5)

Question no – (2)

Solution :

In the question, 3x2 + 17x + 10

∴ 3x2 + 17x + 10

= 3x2 + 15x + 2x + 10

= 3x (x + 5) + 2 (x + 5)

= (3x + 2) (x + 5)

Question no – (3)

Solution :

Given, 5x2 + 9x – 2

∴ 5x2 + 9x – 2

= 5x2 + 10x – x – 2

= 5x (x + 2) – 1 (x + 2)

= (3x – 1) (x + 2)

Question no – (4)

Solution :

From the question we get, 6x2 – 7x – 5

∴ 6x2 – 7x – 5

= 6x2 – 10x + 3x – 5

= 2x (3x – 5) + 1 (3x – 5)

= (3x – 5 ) (2x + 1)

Question no – (5)

Solution :

In the question we get, 6y2 – 17y + 12

∴ 6y2 – 17y + 12

= 6y2 – 9y – 8y + 12

= 3y (2y – 3) – 4 (2y – 3)

= (2y – 3) (3y – 4)

Question no – (6)

Solution :

Given in the question, 8y2 – 2y – 1

∴ 8y2 – 2y – 1

= 8y2 – 4y + 2y – 1

= 4y (2y – 1) + 1 (2y – 1)

= (2y – 1) (4y + 1)

Question no – (7)

Solution :

In the given question, 18b x2 + 18bx – 20b

∴ 18b x2 + 18bx – 20b

= 2b (9x2 + 9x – 10)

= 2b (9x2 + 15x – 6x – 10)

= 2b [3x (2x + 5) – 2 (3x + 5)

= 2b (3x + 5) (3x – 2)

Question no – (8)

Solution :

Given, 14x2 – 60xy + 16y2

Therefore, 14x2 – 60xy + 16y2

= 2 [7x2 – 30xy + 9y2]

= 2 [7x2 – 28xy – 2xy + 8y2]

= 2 [7x2 – 28xy – 2xy + 8y2]

= 2 (x – 4y) (7x – 2y)

Question no – (9)

Solution :

In the given question, 30x2 + 103xy – 7y2

∴ 30x2 + 103xy – 7y2

= 30x2 + 103xy – 2y – 7y2

= 15x (2x + 7y) – y (2x + 7y)

= (15x – y) (2x + 7y)

Question no – (10)

Solution :

Given, 12x2 – 29xy + 14y2

12x2 – 29xy + 14y2

= 12x2 – 8xy – 21xy + 14y2

= 4x (3x – 2y) – 7y (3x – 2y)

= (3x – 2y) (4x – 7y)

Question no – (11)

Solution :

In the given question, 15 + p (7 – 2p)

∴ 15 + p (7 – 2p)

= 15 + 7p – 2p2

= 15 + 10p – 3p – 2p2

= 5 (3 + 2p) – p (3 + 2p)

= (3 + 2p) (5 – p)

Question no – (13)

Solution :

Given, x (2x + 5) – 3

= x (2x + 5) – 3

= 2x2 + 5x – 3 = 2×2 + 6x – x – 3

= 2x (x + 3) – 1 (x + 3)

= (2x – 1) (x + 3)

Question no – (14)

Solution :

In the question we get, 1 – 18y – 63y2

∴ 1 – 18y – 63y2

= 1 – 21y + 3y – 63y2

= 1 (1 – 21y) + 3y (1 – 21y)

= (1 – 21y) (1 + 3y)

Question no – (15)

Solution :

From the question we get, 8a2b – 10a2b2 – 12ab3

∴ 8a2b – 10a2b2 – 12ab3

= 2ab (4a2 – 5ab – 6b2)

= 2ab [4a2 – 8ab + 3ab – 6b2)

= 2ab [4a (a – 2b) + 3b (a – 2b)]

= 2ab (4a + 3b) (a – 2b)

Question no – (16)

Solution :

Given, 2(a + b)2 – 5 (a + b) – 3

Therefore, = 2(a + b)2 – 5(a + b) – 3

= 2x2 – 5x – 3

= 2x2 – 6x + x – 3

= 2x (x – 3) + 1 (x – 3)

= (x – 3) (2x + 1)

= (a + b – 3) (2a + 2b + 1)

Next Chapter Solution :

Updated: June 19, 2023 — 2:54 pm