SELINA Solutions for Class 9 Maths Chapter 5 - Factorisation
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)14y
Question 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 - 3
Solution 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 - 4b
Solution 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 + 8
Solution 7
Question 8
Factorize by the grouping method:
ab - 2b + a2 - 2a
Solution 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 - a
Solution 10
Question 11
Factorize by the grouping method:
(ax + by)2 + (bx - ay)2
Solution 11
Question 12
Factorize by the grouping method:
a2x2 + (ax2 + 1) x + a
Solution 12
Question 13
Factorize by the grouping method:
(2a-b)2 -10a + 5b
Solution 13
Question 14
Factorize by the grouping method:
a (a -4) - a + 4
Solution 14
Question 15
Factorize by the grouping method:
y2 - (a + b) y + ab
Solution 15
Question 16
Factorize by the grouping method:
Solution 16
Question 17
Factorise using the grouping method:
x2 + y2 + x + y + 2xy
Solution 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 - 4ab
Solution 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 - 15y
Solution 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)2
Solution 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 + 24
Solution 1
Question 2
Factorize:
a2 - 3a - 40
Solution 2
Question 3
Factorize:
1 - 2a - 3a2
Solution 3
Question 4
Factorize:
x2 - 3ax - 88a2
Solution 4
Question 5
Factorize:
6a2 - a-15
Solution 5
Question 6
Factorize:
24a3 + 37a2 - 5a
Solution 6
Question 7
Factorize:
a(3a - 2) - 1
Solution 7
Question 8
Factorize:
a2b2 + 8ab - 9
Solution 8
Question 9
Factorize:
3 - a (4 + 7a)
Solution 9
Question 10
Factorize:
(2a + b)2 - 6a - 3b - 4
Solution 10
Question 11
Factorize:
1 - 2 (a+ b) - 3 (a + b)2
Solution 11
Question 12
Factorize:
3a2 - 1 - 2a
Solution 12
Question 13
Factorize:
x2 + 3x + 2 + ax + 2a
Solution 13
Question 14
Factorize:
(3x - 2y)2 + 3 (3x - 2y) - 10
Solution 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 + 25
Solution 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 - 9b2
Solution 1
Question 2
Factorize :
a2 - (2a + 3b)2
Solution 2
Question 3
Factorize :
a2 - 81 (b-c)2
Solution 3
Question 4
Factorize :
25(2a - b)2 - 81b2
Solution 4
Question 5
Factorize :
50a3 - 2a
Solution 5
Question 6
Factorize :
4a2b - 9b3
Solution 6
Question 7
Factorize :
3a5 - 108a3
Solution 7
Question 8
Factorize :
9(a - 2)2 - 16(a + 2)2
Solution 8
Question 9
Factorize :
a4 - 1
Solution 9
Question 10
Factorize :
a3 + 2a2 - a-2
Solution 10
Question 11
Factorize :
(a + b)3 - a - b
Solution 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 - 7b
Solution 14
Question 15
Factorize :
9a2 + 3a - 8b - 64b2
Solution 15
Question 16
Factorize :
4a2 - 12a + 9 - 49b2
Solution 16
Question 17
Factorize :
4xy - x2 - 4y2 + z2
Solution 17
Question 18
Factorize :
a2 + b2 - c2 - d2 + 2ab - 2cd
Solution 18
Question 19
Factorize :
4x2 - 12ax - y2 - z2 - 2yz + 9a2
Solution 19
Question 20
Factorize :
(a2 - 1) (b2 - 1) + 4ab
Solution 20
Question 21
Factorize :
x4 + x2 + 1
Solution 21
Question 22
Factorize :
(a2 + b2 - 4c2)2 - 4a2b2
Solution 22
Question 23
Factorize :
(x2 + 4y2 - 9z2)2 - 16x2y2
Solution 23
Question 24
(a + b) 2 - a2 + b2
Solution 24
Question 25
a2 - b2 - (a + b) 2
Solution 25
Question 26
9a2 - (a2 - 4) 2
Solution 26
Question 27
Solution 27
Question 28
Solution 28
Question 29
4x4 - x2 - 12x - 36
Solution 29
Question 30
a2 ( b + c) - (b + c)3
Solution 30
Chapter 5 - Factorisation Exercise Ex. 5(D)
Question 1
Factorize :
a3 - 27
Solution 1
Question 2
Factorize :
1 - 8a3
Solution 2
Question 3
Factorize :
64 - a3b3
Solution 3
Question 4
Factorize :
a6 + 27b3
Solution 4
Question 5
Factorize :
3x7y - 81x4y4
Solution 5
Question 6
Factorize :
a3 - 
Solution 6
Question 7
Factorize :
a3 + 0.064
Solution 7
Question 8
Factorize :
a4 - 343a
Solution 8
Question 9
Factorise:
(x - y)3 - 8x3
Solution 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 - b6
Solution 11
Question 12
Factorize :
a6 - 7a3 - 8
Solution 12
Question 13
Factorize :
a3 - 27b3 + 2a2b - 6ab2
Solution 13
Question 14
Factorize :
8a3 - b3 - 4ax + 2bx
Solution 14
Question 15
Factorize :
a - b - a3 + b3
Solution 15
Question 16
Factorise:
2x3 + 54y3 - 4x - 12y
Solution 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 - 3x3
Solution 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 62
Solution 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 + 11x2y2
Solution 5
Question 6
Factorize :
Solution 6
Question 7
Factorize :
a - b - 4a2 + 4b2
Solution 7
Question 8
Factorize :
(2a - 3)2 - 2 (2a - 3) (a - 1) + (a - 1)2
Solution 8
Question 9
Factorize :
(a2 - 3a) (a2 + 3a + 7) + 10
Solution 9
Question 10
Factorize :
(a2 - a) (4a2 - 4a - 5) - 6
Solution 10
Question 11
Factorize :
x4 + y4 - 3x2y2
Solution 11
Question 12
Factorize :
5a2 - b2 - 4ab + 7a - 7b
Solution 12
Question 13
Factorize :
12(3x - 2y)2 - 3x + 2y - 1
Solution 13
Question 14
Factorize :
4(2x - 3y)2 - 8x+12y - 3
Solution 14
Question 15
Factorize :
3 - 5x + 5y - 12(x - y)2
Solution 15
Question 16
9x 2 + 3x - 8y - 64y2
Solution 16
Question 17
Solution 17
Question 18
Solution 18
Question 19
2(ab + cd) - a2 - b2 + c2 + d2
Solution 19
Question 20
Solution 20
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