ICSE Class 9 Factorisation Solution New Pattern By Clarify Knowledge

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ICSE Class 9 Factorisation Solution Table

Chapter 5 - Factorisation Exercise Ex. 5(A)

Question 1

Factorise by taking out the common factors:

2 (2x - 5y) (3x + 4y) - 6 (2x - 5y) (x - y)Solution 1

Taking (2x - 5y) common from both terms

= (2x - 5y)[2(3x + 4y) - 6(x - y)]

=(2x - 5y)(6x + 8y - 6x + 6y)

=(2x - 5y)(8y + 6y)

=(2x - 5y)(14y)

=(2x - 5y)14yQuestion 2

Factories by taking out common factors:

xy(3x² - 2y²) - yz(2y² - 3x²) + zx(15x² - 10y²)Solution 2

xy(3x² - 2y²) - yz(2y² - 3x²) + zx(15x² - 10y²)

= xy(3x² - 2y²) + yz(3x² - 2y²) + zx(15x² - 10y²)

= xy(3x² - 2y²) + yz(3x² - 2y²) + 5zx(3x² - 2y²)

= (3x² - 2y²)[xy + yz + 5zx]Question 3

Factories by taking out common factors:

ab(a² + b² - c²) - bc(c² - a² - b²) + ca(a² + b² - c²)Solution 3

ab(a² + b² - c²) - bc(c² - a² - b²) + ca(a² + b² - c²)

= ab(a² + b² - c²) + bc(a² + b² - c²) + ca(a² + b² - c²)

= (a² + b² - c²)[ab + bc + ca]Question 4

Factories by taking out common factors:

2x(a - b) + 3y(5a - 5b) + 4z(2b - 2a)Solution 4

2x(a - b) + 3y(5a - 5b) + 4z(2b - 2a)

= 2x(a - b) + 15y(a - b) - 8z(a - b)

= (a - b)[2x + 15y - 8z]Question 5

Factorize by the grouping method:

a³ + a - 3a² - 3Solution 5

a³ + a - 3a² - 3= a (a² + 1) - 3(a² + 1) 

= (a² + 1) (a -3).Question 6

Factorize by the grouping method:

16 (a + b)² - 4a - 4bSolution 6

16 (a + b)² - 4a - 4b =16 (a + b)² - 4 (a + b)

= 4 (a + b) [4 (a + b) - 1]

= 4 (a + b) (4a + 4b - 1)Question 7

Factorize by the grouping method:

a⁴ - 2a³ - 4a + 8Solution 7

Question 8

Factorize by the grouping method:

ab - 2b + a² - 2aSolution 8

Question 9

Factorize by the grouping method:

ab (x² + 1) + x (a² + b²)Solution 9

Question 10

Factorize by the grouping method:

a² + b - ab - aSolution 10

Question 11

Factorize by the grouping method:

(ax + by)² + (bx - ay)²Solution 11

Question 12

Factorize by the grouping method:

a²x² + (ax² + 1) x + aSolution 12

Question 13

Factorize by the grouping method:

(2a-b)² -10a + 5bSolution 13

Question 14

Factorize by the grouping method:

a (a -4) - a + 4Solution 14

Question 15

Factorize by the grouping method:

y² - (a + b) y + abSolution 15

Question 16

Factorize by the grouping method:

Solution 16

Question 17

Factorise using the grouping method:

x² + y² + x + y + 2xySolution 17

= (x² + y² + 2xy ) + (x + y)

[As (x + y)² = x² + 2xy + y²]

=(x + y)² + (x + y)

=(x + y)(x + y + 1)Question 18

Factorise using the grouping method:

a² + 4b² - 3a + 6b - 4abSolution 18

= a² + 4b² - 4ab - 3a + 6b

= a² + (2b)² - 2 × a × (2b) - 3(a - 2b)

[As (a - b)² = a² - 2ab + b² ]

=(a - 2b)² - 3(a - 2b)

=(a - 2b)[(a - 2b)- 3]

=(a - 2b)(a - 2b - 3)Question 19

Factorise using the grouping method:

m (x - 3y)² + n (3y - x) + 5x - 15ySolution 19

= m (x - 3y)² - n (x - 3y) + 5(x - 3y)

[Taking (x - 3y) common from all the three terms]

=(x - 3y) [m(x - 3y) - n + 5]

=(x - 3y)(mx - 3my - n + 5)Question 20

Factorise using the grouping method:

x (6x - 5y) - 4 (6x - 5y)²Solution 20

=(6x - 5y)[x - 4(6x - 5y)]

[Taking (6x - 5y) common from the three terms]

= (6x - 5y)(x - 24x + 20y)

= (6x - 5y)(-23x + 20y)

= (6x - 5y)(20y - 23x)

Chapter 5 - Factorisation Exercise Ex. 5(B)

Question 1

Factorize:

a² + 10a + 24Solution 1

Question 2

Factorize:

a² - 3a - 40Solution 2

Question 3

Factorize:

1 - 2a - 3a²Solution 3

Question 4

Factorize:

x² - 3ax - 88a²Solution 4

Question 5

Factorize:

6a² - a-15Solution 5

Question 6

Factorize:

24a³ + 37a² - 5aSolution 6

Question 7

Factorize:

a(3a - 2) - 1Solution 7

Question 8

Factorize:

a²b² + 8ab - 9Solution 8

Question 9

Factorize:

3 - a (4 + 7a)Solution 9

Question 10

Factorize:

(2a + b)² - 6a - 3b - 4Solution 10

Question 11

Factorize:

1 - 2 (a+ b) - 3 (a + b)²Solution 11

Question 12

Factorize:

3a² - 1 - 2aSolution 12

Question 13

Factorize:

x² + 3x + 2 + ax + 2aSolution 13

Question 14

Factorize:

(3x - 2y)² + 3 (3x - 2y) - 10Solution 14

Question 15

Factorize:

5 - (3a² - 2a) (6 - 3a² + 2a)Solution 15

Question 16

Solution 16

Question 17

Factories: (x² - 3x)(x² - 3x - 1) - 20.Solution 17

(x² - 3x)(x² - 3x - 1) - 20

= (x² - 3x)[(x² - 3x) - 1] - 20

= a[a - 1] - 20 ….(Taking x² - 3x = a)

= a² - a - 20

= a² - 5a + 4a - 20

= a(a - 5) + 4(a - 5)

= (a - 5)(a + 4)

= (x² - 3x - 5)(x² - 3x + 4)Question 18

Find each trinomial (quadratic expression), given below, find whether it is factorisable or not. Factorise, if possible.

(i) x² - 3x - 54

(ii) 2x² - 7x - 15

(iii) 2x² + 2x - 75

(iv) 3x² + 4x - 10

(v) x(2x - 1) - 1 Solution 18

Question 19

Solution 19

Question 20

Give possible expressions for the length and the breadth of the rectangle whose area is

12x² - 35x + 25Solution 20

12x² - 35x + 25

= 12x² - 20x - 15x + 25

= 4x(3x - 5) - 5(3x - 5)

= (3x - 5)(4x - 5)

Thus,

Length = (3x - 5) and breadth = (4x - 5)

OR

Length = (4x - 5) and breadth = (3x - 5)

Chapter 5 - Factorisation Exercise Ex. 5(C)

Question 1

Factorize :

25a² - 9b²Solution 1

Question 2

Factorize :

a² - (2a + 3b)²Solution 2

Question 3

Factorize :

a² - 81 (b-c)²Solution 3

Question 4

Factorize :

25(2a - b)² - 81b²Solution 4

Question 5

Factorize :

50a³ - 2aSolution 5

Question 6

Factorize :

4a²b - 9b³Solution 6

Question 7

Factorize :

3a⁵ - 108a³Solution 7

Question 8

Factorize :

9(a - 2)² - 16(a + 2)²Solution 8

Question 9

Factorize :

a⁴ - 1Solution 9

Question 10

Factorize :

a³ + 2a² - a-2Solution 10

Question 11

Factorize :

(a + b)³ - a - bSolution 11

Question 12

Factorize :

a (a - 1) - b (b - 1)Solution 12

Question 13

Factorize :

4a² - (4b² + 4bc + c²)Solution 13

Question 14

Factorize :

4a² - 49b² + 2a - 7bSolution 14

Question 15

Factorize :

9a² + 3a - 8b - 64b²Solution 15

Question 16

Factorize :

4a² - 12a + 9 - 49b²Solution 16

Question 17

Factorize :

4xy - x² - 4y² + z²Solution 17

Question 18

Factorize :

a² + b² - c² - d² + 2ab - 2cdSolution 18

Question 19

Factorize :

4x² - 12ax - y² - z² - 2yz + 9a²Solution 19

Question 20

Factorize :

(a² - 1) (b² - 1) + 4abSolution 20

Question 21

Factorize :

x⁴ + x² + 1Solution 21

Question 22

Factorize :

(a² + b² - 4c²)² - 4a²b²Solution 22

Question 23

Factorize :

(x² + 4y² - 9z²)² - 16x²y²Solution 23

Question 24

(a + b) ² - a² + b²Solution 24

Question 25

a² - b² - (a + b) ²Solution 25

Question 26

9a² - (a² - 4) ²Solution 26

Question 27

Solution 27

Question 28

Solution 28

Question 29

4x⁴ - x² - 12x - 36Solution 29

Question 30

a² ( b + c) - (b + c)³Solution 30

Chapter 5 - Factorisation Exercise Ex. 5(D)

Question 1

Factorize :

a³ - 27Solution 1

Question 2

Factorize :

1 - 8a³Solution 2

Question 3

Factorize :

64 - a³b³Solution 3

Question 4

Factorize :

a⁶ + 27b³Solution 4

Question 5

Factorize :

3x⁷y - 81x⁴y⁴Solution 5

Question 6

Factorize :

a³ - Solution 6

Question 7

Factorize :

a³ + 0.064Solution 7

Question 8

Factorize :

a⁴ - 343aSolution 8

Question 9

Factorise:

(x - y)³ - 8x³Solution 9

= (x - y)³ - (2x)³

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

[Using identity (a³ - b³) = (a - b)(a² + ab + b²)]

= (-x - y)[x² + y² - 2xy + 2x² - 2xy + 4x²]

=-(x + y) [7x² - 4xy + y²]Question 10

Factorize :

Solution 10

Question 11

Factorize :

a⁶ - b⁶Solution 11

Question 12

Factorize :

a⁶ - 7a³ - 8Solution 12

Question 13

Factorize :

a³ - 27b³ + 2a²b - 6ab²Solution 13

Question 14

Factorize :

8a³ - b³ - 4ax + 2bxSolution 14

Question 15

Factorize :

a - b - a³ + b³Solution 15

Question 16

Factorise:

2x³ + 54y³ - 4x - 12ySolution 16

= 2(x³ + 27y³ - 2x - 6y)

= 2{[(x)³+(3y)³] - 2(x  + 3y)}

[Using identity (a³ +  b³) = (a + b)(a² - ab + b²)]

=2{[(x + 3y)(x² - 3xy + 9y²)] - 2(x + 3y)}

=2(x + 3y)(x² - 3xy + 9y² - 2)Question 17

1029 - 3x³Solution 17

1029 - 3x³

= 3(343 - x³)

= 3(7³ - x³)

= 3(7 - x)(7² + 7x + x²)

= 3(7 - x)(49 + 7x + x²)Question 18

Show that:

(i) 13³ - 5³ is divisible by 8

(ii)35³ + 27³ is divisible by 62Solution 18

(i) (13³ - 5³)

[Using identity (a³ - b³) = (a - b)(a² + ab + b²)]

=(13 - 5)(13² + 13 × 5 + 5²)

=8(169 + 65 + 25)

Therefore, the number is divisible by 8.

(ii) (35³ + 27³)

[Using identity (a³ + b³)=(a + b)(a² - ab + b²)]

=(35 + 27)(35² + 35× 27 + 27²)

=62 × (35² + 35 × 27 + 27²)

Therefore, the number is divisible by 62.Question 19

Solution 19

Chapter 5 - Factorisation Exercise Ex. 5(E)

Question 1

Factorize :

Solution 1

Question 2

Factorize :

Solution 2

Question 3

Factorize :

Solution 3

Question 4

Factorize :

Solution 4

Question 5

Factorize :

4x⁴ + 9y⁴ + 11x²y²Solution 5

Question 6

Factorize :

Solution 6

Question 7

Factorize :

a - b - 4a² + 4b²Solution 7

Question 8

Factorize :

(2a - 3)² - 2 (2a - 3) (a - 1) + (a - 1)²Solution 8

Question 9

Factorize :

(a² - 3a) (a² + 3a + 7) + 10Solution 9

Question 10

Factorize :

(a² - a) (4a² - 4a - 5) - 6Solution 10

Question 11

Factorize :

x⁴ + y⁴ - 3x²y²Solution 11

Question 12

Factorize :

5a² - b² - 4ab + 7a - 7bSolution 12

Question 13

Factorize :

12(3x - 2y)² - 3x + 2y - 1Solution 13

Question 14

Factorize :

4(2x - 3y)² - 8x+12y - 3Solution 14

Question 15

Factorize :

3 - 5x + 5y - 12(x - y)²Solution 15

Question 16

9x ² + 3x - 8y - 64y²Solution 16

Question 17

Solution 17

Question 18

Solution 18

Question 19

2(ab + cd) - a² - b² + c² + d²Solution 19

Question 20

Solution 20