ICSE Solutions for Selina Concise Chapter 5 Factorisation Class 9 Maths
Exercise 5(A)
2 (2x – 5y) (3x + 4y) – 6 (2x – 5y) (x – y)
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
xy(3x2 – 2y2) – yz(2y2 – 3x2) + zx(15x2 – 10y2)
Answer
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)
3. Factories by taking out common factors :
ab(a2 + b2 – c2) – bc(c2 – a2 – b2) + ca(a2 + b2 – c2)
Answer
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)
4. Factories by taking out common factors :
2x(a – b) + 3y(5a – 5b) + 4z(2b – 2a)
2x(a – b) + 3y(5a – 5b) + 4z(2b – 2a)
= 2x(a – b) + 15y(a – b) – 8z(a – b)
= (a – b)(2x + 15y – 8z)
5. Factorise by the grouping method : a3 + a – 3a2 – 3
Answer
a3 + a – 3a2 – 3
= a (a2 + 1) – 3(a2 + 1)
= (a2 + 1) (a -3)
6. Factorise by the grouping method: 16 (a + b)2 – 4a – 4b
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)
7. Factorise by the grouping method : a4 – 2a3 – 4a + 8
Answer
a4 – 2a3 – 4a + 8 = a3(a – 2) – 4(a – 2)
= (a3 – 4)(a – 2)
8. Factorise by the grouping method : ab – 2b + a2 – 2a
Answer
ab – 2b + a2 – 2a = b(a – 2) + a(a – 2)
= (a + b)(a – 2)
9. Factorise by the grouping method : ab (x2 + 1) + x (a2 + b2)
Answer
ab (x2 + 1) + x (a2 + b2) = abx2 + ab + a2x + b2x
= ax(bx + a) + b(bx + a)
= (ax + b)(bx + a)
10. Factorise by the grouping method : a2 + b – ab – a
Answer
a2 + b – ab – a = a2 – a + b – ab
= a(a – 1) + b(1 – a)
= a(a – 1) – b(a – 1)
= (a -1)(a – b)
11. Factorise by the grouping method : (ax + by)2 + (bx – ay)2
Answer
(ax + by)2 + (bx – ay)2
= a2x2 + b2y2 + 2axby + b2x2 + a2y2 – 2bxay
= a2x2 + b2y2 + b2x2 + a2y2
= x2(a2 + b2) + y2(a2 + b2)
= (x2 + y2)(a2 + b2)
12. Factorise by the grouping method : a2x2 + (ax2 + 1) x + a
Answer
a2x2 + (ax2 + 1) x + a
= a2x2 + a + (ax2 + 1) x
= a(ax2 + 1) + x( ax2 + 1)
= (a + x)(ax2 + 1)
13. Factorise by the grouping method : (2a-b)2 -10a + 5b
Answer
(2a – b)2 – 10a + 5b
= (2a – b)2 – 5(2a – b)
= (2a – b)(2a – b – 5)
14. Factorise by the grouping method : a (a -4) – a + 4
Answer
a (a -4) – a + 4
= a(a – 4) -1(a – 4)
= (a – 4 )( a – 1)
15. Factorise by the grouping method : y2 – (a + b) y + ab
Answer
y2 – (a + b) y + ab
= y2 – ay – by + ab
= y(y – a) – b(y – a)
= (y – a)(y – b)
16. Factorise by the grouping method :
Answer
17. Factorise using the grouping method:
x2 + y2 + x + y + 2xy
Answer
x2 + y2 + x + y + 2xy
= (x2 + y2 + 2xy) + (x + y) [As (x + y)2 = x2 + 2xy + y2]
and = (x + y)2 + (x + y)
= (x + y)(x + y + 1)
18. Factorise using the grouping method :
a2 + 4b2 – 3a + 6b – 4ab
Answer
a2 + 4b2 – 3a + 6b – 4ab
= 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)
19. Factorise using the grouping method :
m (x – 3y)2 + n (3y – x) + 5x – 15y
Answer
m (x – 3y)2 + n (3y – x) + 5x – 15y
= 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)
20. Factorise using the grouping method :
x (6x – 5y) – 4 (6x – 5y)2
Answer
x (6x – 5y) – 4 (6x – 5y)2
= (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)
Exercise 5(B)
1. Factorise : a2 + 10a + 24
Answer
a2 + 10a + 24
= a2 + 6a + 4a + 24
= a(a + 6) + 4(a + 6)
= (a + 6)(a + 4)
2. Factorise : a2 – 3a – 40
Answer
a2 – 3a – 40
= a2 – 8a + 5a – 40
= a(a – 8) + 5(a – 8)
= (a – 8)(a + 5)
3. Factorise : 1 – 2a – 3a2
Answer
1 – 2a – 3a2
= 1 – 3a + a – 3a2
=(1 + a)(1 – 3a)
4. Factorise : x2 – 3ax – 88a2
Answer
x2 – 3ax – 88a2
= x2 – 11ax + 8ax – 88a2
= x(x – 11a) + 8a(x – 11a)
= (x + 8a)(x – 11a)
5. Factorise : 6a2 – a – 15
Answer
6a2 – a – 15
= 6a2 – 10a + 9a – 15
= 2a(3a – 5) + 3(3a – 5)
= (2a + 3)(3a – 5)
6. Factorise : 24a3 + 37a2 – 5a
Answer
24a3 + 37a2 – 5a
= a(24a2 + 37a – 5)
= a(24a2 + 40a -3a – 5)
= a[ 8a(3a + 5) – 1(3a + 5)]
= a[(8a – 1)(3a + 5)]
= a(8a – 1)(3a + 5)
7. Factorise : a(3a – 2) – 1
Answer
a(3a – 2) – 1
= 3a2 – 2a – 1
= 3a2 – 3a + a – 1
= 3a(a – 1) + 1(a – 1)
= (3a + 1)(a – 1)
8. Factorise : a2b2 + 8ab – 9
Answer
a2b2 + 8ab – 9
= a2b2 + 9ab – ab – 9
= ab(ab + 9) -1(ab + 9)
= (ab + 9)(ab – 1)
9. Factorise : 3 – a (4 + 7a)
Answer
3 – a (4 + 7a)
= 3 – 4a – 7a2
= 3 – 7a + 3a – 7a2
= 1(3 – 7a) + a(3 – 7a)
= (3 – 7a)(a + 1)
10. Factorise : (2a + b)2 – 6a – 3b – 4
Answer
(2a + b)2 – 6a – 3b – 4
= (2a + b)2 – 3(2a + b) – 4
Assume that, 2a + b = x
Therefore,
(2a + b)2 – 6a – 3b – 4
= x2 – 3x – 4
= x2 – 4x + x – 4
= 1(x – 4) + x(x – 4)
= (x + 1)(x – 4)
= (2a + b + 1)(2a + b – 4) [resubstitute the value of x]
11. Factorise : 1 – 2 (a+ b) – 3 (a + b)2
Answer
1 – 2 (a+ b) – 3 (a + b)2
Assume that a + b = x ;
1 – 2(a + b) – 3(a + b)2
= 1 – 2x – 3×2
= 1 – 3x + x – 3×2
= 1(1 – 3x) + x(1 – 3x)
= (1 – 3x)(1 + x)
= [1 – 3(a + b)][1 + (a + b)]
= (1 – 3a – 3b)(1 + a + b)
12. Factorise : 3a2 – 1 – 2a
Answer
3a2 – 1 – 2a
= 3a2 – 2a – 1
= 3a2 – 3a + a – 1
= 3a(a – 1) + 1(a – 1)
= (3a + 1)(a – 1)
13. Factorise : x2 + 3x + 2 + ax + 2a
Answer
x2 + 3x + 2 + ax + 2a
= x2 + 2x + x + 2 + ax + 2a
= x(x + 2)+ 1(x + 2) + a(x + 2)
= (x + 2)(x + a + 1)
14. Factorise : (3x – 2y)2 + 3 (3x – 2y) – 10
Answer
(3x – 2y)2 + 3 (3x – 2y) – 10
Assume that 3x – 2y = a
Therefore,
(3x – 2y)2 + 3 (3x – 2y) – 10
= a2 + 3a – 10
= a2 + 5a – 2a -10
= a(a + 5) -2(a + 5)
= (a + 5)(a – 2)
= (3x – 2y + 5)(3x – 2y – 2)
15. Factorise : 5 – (3a2 – 2a) (6 – 3a2 + 2a)
Answer
5 – (3a2 – 2a) (6 – 3a2 + 2a)
= 5 – (3a2 – 2a)[ 6 – (3a2 – 2a)]
Assume that 3a2 – 2a = x
Therefore,
5 – (3a2 – 2a)(6 – 3a2 + 2a)
= 5 – x(6 – x)
= 5 – 6x + x + x2
= 5(1 – x) – x(1 – x)
= (5 – x)(1 – x)
= (x – 5)(x – 1)
= (3a2 – 2a – 5)(3a2 – 2a – 1)
= (3a2 – 5a + 3a -5)(3a2 – 3a + a – 1)
= [ a(3a – 5) + 1(3a – 5)][3a(a – 1) + 1(a – 1)]
= (3a – 5)(a + 1)(3a + 1)(a – 1)
16. Factorise : 1/35 + 12a/35 + a2
Answer
17. Factories: (x2 – 3x)(x2 – 3x – 1) – 20.
Answer
(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)
18.1. Find trinomial (quadratic expression), given below, find whether it is factorisable or not. Factorise, if possible.
x2 – 3x – 54
Answer
Given expression : x2 – 3x – 54
Comparing with ax2 + bx + c, we get a = 1, b = -3, and c = – 54
∴ b2 – 4ac = (-3)2 – 4(1)(-54) = 9 + 216 = 225, which is a perfect square.
∴ x2 – 3x – 54 is factorisable.
Now, x2 – 3x – 54 = x2 – 9x + 6x – 54
= x( x – 9 ) + 6( x – 9 )
= ( x – 9 )( x + 6 )
18.2. Find trinomial (quadratic expression), given below, find whether it is factorisable or not. Factorise, if possible.
2x2 – 7x – 15
Answer
Given expression : 2x2 – 7x – 15
Comparing with ax2 + bx + c, we get a = 2, b = -7, and c = -15
∴ b2 – 4ac = (-7)2 – 4(2)(-15) = 49 + 120 = 169, which is a perfect square.
∴ 2x2 – 7x – 15 is factorisable.
Now, 2x2 – 7x – 15
= 2×2 – 10x + 3x – 15
= 2x( x – 5 ) + 3( x – 5 )
= ( 2x + 3 )( x – 5 )
18.3. Find trinomial (quadratic expression), given below, find whether it is factorisable or not. Factorise, if possible.
2x2 + 2x – 75
Answer
Given expression : 2x2 + 2x – 75
Comparing with ax2 + bx + c, we get a = 2, b = 2, and c = – 75
∴ b2 – 4ac = (2)2 – 4(2)(-75) = 4 + 600 = 604, which is not a perfect square.
∴ 2x2 + 2x – 75 is not factorisable.
18.4. Find trinomial (quadratic expression), given below, find whether it is factorisable or not. Factorise, if possible.
3x2 + 4x – 10
Answer
Given expression : 3x2 + 4x – 10
Comparing with ax2 + bx + c, we get a = 3, b = 4, and c = -10
∴ b2 – 4ac = (4)2 – 4(3)(-10) = 16 + 120 = 136, which is not a perfect square.
∴ 3x2 + 4x – 10 is not factorisable.
18.5. Find trinomial (quadratic expression), given below, find whether it is factorisable or not. Factorise, if possible.
x(2x – 1) – 1
Answer
Given expression : x(2x – 1) – 1
Now , x(2x – 1) – 1 = 2x2 – x – 1
Comparing with ax2 + bx + c, we get a = 2, b = – 1, and c = – 1
∴ b2 – 4ac = (-1)2 – 4(2)(-1) = 1 + 8 = 9, which is a perfect square.
∴ 2x2 – x – 1 is factorisable.
Now, 2x2 – x – 1 = 2×2 – 2x + x – 1
= 2x(x – 1) + 1(x – 1)
= (2x + 1)(x – 1)
19.1. Factorise : 4√3x2 + 5x – 2√3
Answer
4√3x2 + 5x – 2√3
= 4√3x2 + 8x – 3x – 2√3
= 4x(√3x + 2) – √3(√3x + 2)
= (√3x + 2)(4x – √3)
19.2. Factorise : 7√2x2 – 10x – 4√2
Answer
7√2x2 – 10x – 4√2
= 7√2x2 – 14x + 4x – 4√2
= 7√2x(x – √2) + 4(x – √2)
= (x – √2)(7√2x + 4)
20. Give possible expressions for the length and the breadth of the rectangle whose area is 12x2 – 35x + 25
Answer
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)
Exercise 5(C)
1. Factorise : 25a2 – 9b2
Answer
25a2 – 9b2
= (5a)2 – (3b)2
= (5a – 3b)(5a + 3b)
2. Factorise : a2 – (2a + 3b)2
Answer
a2 – (2a + 3b)2
= a2 – (2a + 3b)2
= (a – 2a – 3b)(a + 2a + 3b) [ ∵ a2 – b2 = (a + b)(a – b)]
= (–a – 3b)(3a + 3b)
= -3(a + 3b)(a + b)
3. Factorise : a2 – 81 (b-c)2
Answer
a2 – 81 (b-c)2
= (a)2 – [9(b – c)]2
= [a – (9b – 9c)][a + (9b – 9c)] [∵ a2 – b2 = (a + b)(a – b)]
= (a – 9b + 9c)(a + 9b – 9c)
4. Factorise : 25(2a – b)2 – 81b2
Answer
25(2a – b)2 – 81b2
= [5(2a – b)]2 – (9b)2
= [5(2a – b) – 9b][5(2a – b) + 9b] [∵ a2 – b2 = (a + b)(a – b)]
= [10a – 5b – 9b][10a – 5b + 9b]
= [10a – 14b][10a + 4b]
= 2×(5a – 7b)×2×(5a + 2b)
= 4(5a – 7b)(5a + 2b)
5. Factorise : 50a3 – 2a
Answer
50a3 – 2a
= 2a(25a2 – 1)
= 2a[(5a)2 – (1)2]
= 2a(5a + 1)(5a – 1) [∵ a2 – b2 = (a + b)(a – b)]
6. Factorise : 4a2b – 9b3
Answer
4a2b – 9b3
= b(4a2 – 9b2)
= b[(2a)2 – (3b)2]
= b(2a – 3b)(2a + 3b) [∵ a2 – b2 = (a + b)(a – b)]
7. Factorise : 3a5 – 108a3
Answer
3a5 – 108a3
= 3a3(a2 – 36)
= 3a3[a2 – 62]
= 3a3(a – 6)(a + 6) [∵ a2 – b2 = (a + b)(a – b)]
8. Factorise : 9(a – 2)2 – 16(a + 2)2
Answer
9(a – 2)2 – 16(a + 2)2
= [3(a – 2)]2 – [4(a + 2)]2
= [ 3(a – 2) – 4(a + 2)][3(a – 2) + 4(a + 2)]
[∵ a2 – b2 = (a + b)(a – b)]
= [3a – 6 – 4a – 8][3a – 6 + 4a + 8]
= (–a – 14)(7a + 2)
= –(a + 14)(7a + 2)
9. Factorise : a4 – 1
Answer
a4 – 1
= (a2)2 – (1)2
= (a2 + 1)(a2 – 1) [∵ a2 – b2 = (a + b)(a – b)]
= (a2 + 1)[(a)2 – (1)2]
= (a2 + 1)(a + 1)(a – 1)
10. Factorise : a3 + 2a2 – a – 2
Answer
a3 + 2a2 – a – 2
= a2(a + 2) – 1(a + 2)
= (a2 – 1)(a + 2)
= (a + 1)(a – 1)(a + 2) [∵ a2 – b2 = (a + b)(a – b)]
11. Factorise : (a + b)3 – a – b
Answer
(a + b)3 – a – b
= (a + b)3 – (a + b)
= (a + b)[(a + b)2 – 1]
= (a + b)[(a + b)2 – (1)2]
= (a + b)[(a + b) + 1][(a + b) – 1] [∵ a2 – b2 = (a + b)(a – b)]
= (a + b)(a + b + 1)(a + b – 1)
12. Factorise : a (a – 1) – b (b – 1)
Answer
a (a – 1) – b (b – 1)
= a2 – a – b2 + b
= a2 – b2 – a + b
= (a + b)(a – b) – (a – b) [∵ a2 – b2 = (a + b)(a – b)]
= (a – b)[(a + b) – 1]
= (a – b)[a + b – 1]
13. Factorise : 4a2 – (4b2 + 4bc + c2)
Answer
4a2 – (4b2 + 4bc + c2)
= (2a)2 – (2b + c)2
= [2a – (2b + c)][2a + (2b + c)] [∵ a2 – b2 = (a + b)(a – b)]
= [ 2a – 2b – c ][ 2a + 2b + c ]
14. Factorise : 4a2 – 49b2 + 2a – 7b
Answer
4a2 – 49b2 + 2a – 7b
= [(2a)2 – (7b)2] + [2a – 7b]
= [2a – 7b][2a + 7b] + [2a – 7b] [∵ a2 – b2 = (a + b)(a – b)]
= [2a – 7b][2a + 7b + 1]
15. Factorise : 9a2 + 3a – 8b – 64b2
Answer
9a2 + 3a – 8b – 64b2
= 9a2 – 64b2 + 3a – 8b
= (3a)2 – (8b)2 + 3a – 8b
= (3a – 8b)(3a + 8b) + (3a – 8b) [∵ a2 – b2 = (a + b)(a – b)]
= (3a – 8b)(3a + 8b + 1)
16. Factorise : 4a2 – 12a + 9 – 49b2
Answer
4a2 – 12a + 9 – 49b2
= (2a)2 – 12a + (3)2 – 49b2
= (2a – 3)2 – 49b2
= (2a – 3)2 – (7b)2
= (2a – 3 – 7b)(2a – 3 + 7b) [∵ a2 – b2 = (a + b)(a – b)]
17. Factorise : 4xy – x2 – 4y2 + z2
Answer
4xy – x2 – 4y2 + z2
= z2 – (x2 + 4y2 – 4xy)
= z2 – (x – 2y)2
= [z – (x – 2y)][z + (x – 2y)] [∵ a2 – b2 = (a + b)(a – b)]
= [z – x + 2y][z + x – 2y]
18. Factorise : a2 + b2 – c2 – d2 + 2ab – 2cd
Answer
a2 + b2 – c2 – d2 + 2ab – 2cd
= (a2 + b2 + 2ab) – ( c2 + d2 + 2cd )
= (a + b)2 – (c + d)2
= [(a + b) – (c + d)][(a + b) + (c + d)] [∵ a2 – b2 = (a + b)(a – b)]
= (a + b – c – d)(a + b + c + d)
19. Factorise : 4x2 – 12ax – y2 – z2 – 2yz + 9a2
Answer
4x2 – 12ax – y2 – z2 – 2yz + 9a2
= 4x2 + 9a2 – 12ax – y2 – z2 – 2yz
= (2x)2 + (3a)2 – 12ax – (y2 + z2 + 2yz)
= (2x – 3a)2 – (y + z)2
= [(2x – 3a) – (y + z)][(2x – 3a) + (y + z)] [∵ a2 – b2 = (a + b)(a – b)]
= [2x – 3a – y – z][2x – 3a + y + z]
20. Factorise : (a2 – 1) (b2 – 1) + 4ab
Answer
(a2 – 1) (b2 – 1) + 4ab
= a2b2 – a2 – b2 + 1 + 4ab
= a2b2 + 1 + 2ab – a2 – b2 + 2ab
= (a2b2 + 1 + 2ab) – (a2 + b2 – 2ab)
= (ab + 1)2 – (a – b)2
= [(ab + 1) – (a – b)][(ab + 1) + (a – b)] [∵ a2 – b2 = (a + b)(a – b)]
= [ab + 1 – a + b][ab + 1 + a – b]
21. Factorise : x4 + x2 + 1
Answer
x4 + x2 + 1
= x4 + 2x2 + 1 – x2
= (x2)2 + 2x2 + (1)2 – x2
= (x2 + 1)2 – (x)2 [∵ a2 – b2 = (a + b)(a – b)]
= (x2 + 1 – x)(x2 + 1 + x)
22. Factorise : (a2 + b2 – 4c2)2 – 4a2b2
Answer
(a2 + b2 – 4c2)2 – 4a2b2
= (a2 + b2 – 4c2)2 – (2ab)2
= (a2 + b2 – 4c2 – 2ab)(a2 + b2 – 4c2 + 2ab) [∵ a2 – b2 = (a + b)(a – b)]
= (a2 + b2 – 2ab – 4c2)(a2 + b2 + 2ab – 4c2)
= [(a – b)2 – (2c)2 ][(a + b)2 – (2c)2]
= (a – b + 2c)(a – b – 2c)(a + b + 2c)(a + b – 2c)
23. Factorise : (x2 + 4y2 – 9z2)2 – 16x2y2
Answer
(x2 + 4y2 – 9z2)2 – 16x2y2
= (x2 + 4y2 – 9z2)2 – (4xy)2
= (x2 + 4y2 – 9z2 – 4xy)(x2 + 4y2 – 9z2 + 4xy) [∵ a2 – b2 = (a + b)(a – b)]
= (x2 + 4y2 – 4xy – 9z2)(x2 + 4y2 + 4xy – 9z2)
= [(x – 2y)2 – (3z)2][(x + 2y)2 – (3z)2]
= [(x – 2y) – 3z][(x – 2y) + 3z][(x + 2y) – 3z][(x + 2y) + 3z]
= [x – 2y – 3z][x – 2y + 3z][x + 2y – 3z][x + 2y + 3z]
24. Factorise : (a + b) 2 – a2 + b2
Answer
(a + b) 2 – a2 + b2
= a2 + 2ab + b2 – a2 + b2
= 2ab + 2b2
= 2b(a + b)
25. Factorise : a2 – b2 – (a + b) 2
Answer
a2 – b2 – (a + b) 2
= a2 – b2 – (a2 + 2ab + b2)
= a2 – b2 – a2 – 2ab – b2
= –2ab – 2b2
= –2b(a + b)
26. Factorize : 9a2 – (a2 – 4) 2
Answer
9a2 – (a2 – 4) 2
= (3a)2 – (a2 – 4)2
= [3a – (a2 – 4)][3a + (a2 – 4)]
= [3a – a2 – 4][3a + a2 – 4]
= [-a2 + 3a – 4][a2 + 3a – 4]
= [-a2 + 4a – a – 4][a2 + 4a – a – 4]
= [a(-a + 4) + 1(-a + 4)][a(a + 4) – 1(a + 4)]
= [(a + 1)(4 – a)][(a + 4)(a – 1)]
= (a + 1)(4 – a)(a + 4)(a – 1)
27. Factorise : x2 + 1/x2 − 11
Answer
28. Factorise : 4x2 + 1/4x2 + 1
Answer
29. Factorise : 4x4 – x2 – 12x – 36
Answer
Factorise : 4x4 – x2 – 12x – 36
= 4x4 – (x2 + 12x + 36)
= (2x2)2 – (x2 + 2×x×6 + 62)
= (2x2)2 – (x + 6)2
= (2x2 + x + 6)(2x2 – x – 6)
= (2x2 + x + 6)(2x2 – 4x + 3x – 6)
= (2x2 + x + 6)[2x(x – 2) + 3(x – 2)]
= (2x2 + x + 6)[(x – 2)(2x + 3)]
= (2x2 + x + 6)(x – 2)(2x + 3)
30. Factorise : a2 (b + c) – (b + c)3
Answer
a2 (b + c) – (b + c)3
= (b + c)[a2– (b + c)2]
= (b + c)[(a + b + c)(a – b – c)]
= (b + c)(a + b + c)(a – b – c)
Exercise 5(D)
1. Factorise : a3 – 27
Answer
a3 – 27
= (a)3 – (3)3
= (a – 3)[ (a)2 + a×3 + (3)2] [∵ a3 – b3 = (a – b)(a2 + ab + b2)]
= (a – 3)[a2 + 3a + 9]
2. Factorise : 1 – 8a3
Answer
1 – 8a3
= (1)3 – (2a)3
= (1 – 2a)[(1)2 + 1×2a + (2a)2] [∵ a3 – b3 = (a – b)(a2 + ab + b2)]
= (1 – 2a)[1 + 2a + 4a2]
3. Factorise : 64 – a3b3
Answer
64 – a3b3
= (4)3 – (ab)3
= (4 – ab)[(4)2 + 4×ab + (ab)2] [∵ a3 – b3 = (a – b)(a2 + ab + b2)]
= (4 – ab)(16 + 4ab + a2b2)
4. Factorise : a6 + 27b3
Answer
a6 + 27b3
= (a2)3 + (3b)3
= (a2 + 3b)[(a2)2 – a2×3b + (3b)2] [∵ a3 + b3 = (a + b)(a2 – ab + b2)]
= (a2 + 3b)[a4 – 3a2b + 9b2]
5. Factorise : 3x7y – 81x4y4
Answer
3x7y – 81x4y4
= 3xy(x6 – 27x3y3)
= 3xy[(x2)3 – (3xy )3]
= 3xy(x2 – 3xy)[(x2)2 + x2×3xy + (3xy)2] [∵ a3 – b3 = (a -b)(a2 + ab + b2)]
= 3xy(x2 – 3xy)[x4 + 3x3y + 9x2y2]
= 3xy [x(x + 3y) x2(x2 + 3xy + 9y2)]
= 3x4y(x – 3y)(x2 + 3xy + 9y2)
6. Factorise : a3 – 27/a3
Answer
7. Factorise : a3 + 0.064
Answer
a3 + 0.064
= (a)3 + (0.4)3
= (a + 0.4)[(a)2 – a×0.4 + (0.4)2] [∵ a3 + b3 = (a + b)(a2 – ab + b2)]
= (a + 0.4)(a2 – 0.4a + 0.16)
8. Factorise : a4 – 343a
Answer
a4 – 343a
= a(a3 – 73)
= a(a – 7)[(a)2 + a×7 + (7)2] [∵ a3 – b3 = (a – b)(a2 + ab + b2)]
= a(a – 7)(a2 + 7a + 49)
9. Factorise: (x – y)3 – 8x3
Answer
(x – y)3 – 8x3
= (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]
10. Factorise : 8a3/27 – b3/8
Answer
11. Factorise : a6 – b6
Answer
We know that,
a3 + b3 = (a + b)(a2 – ab + b2) ….(1)
a3 – b3 = (a – b)(a2 + ab + b2) ….(2)
a6 – b6
= (a3)2 – (b3)2
= (a3 + b3)(a3 – b3)
= (a + b)(a2 – ab + b2)(a – b)(a2 + ab + b2) [From(1) and (2)]
= (a + b)(a – b)(a2 – ab + b2)(a2 + ab + b2)
12. Factorise : a6 – 7a3 – 8
Answer
We know that,
a3 + b3 = (a + b)(a2 – ab + b2) ….(1)
a3 – b3 = (a – b)(a2 + ab + b2) ….(2)
a6 – 7a3 – 8
= a6 – 8a3 + a3 – 8
= a3(a3 – 8) + 1(a3 – 8)
= (a3 + 1)(a3 – 8)
= (a3 + 13)(a3 – 23)
= (a + 1)(a2 – a + 1)(a – 2)(a2 + 2a + 4)
[From(1) and (2)]
= (a + 1)(a – 2)(a2 – a + 1)(a2 + 2a + 4)
13. Factorise : a3 – 27b3 + 2a2b – 6ab2
Answer
a3 – 27b3 + 2a2b – 6ab2
We know that,
a3 – b3 = (a – b)(a2 + ab + b2) ….(1)
a3 – 27b3 + 2a2b – 6ab2
= (a)3 – (3b)3 + 2ab(a – 3b)
= (a – 3b)[ a2 + a×3b + (3b)2] + 2ab(a – 3b) [From(1)]
= (a – 3b)[a2 + 3ab + 9b2] + 2ab(a – 3b)
= (a – 3b)[a2 + 3ab + 9b2 + 2ab]
= (a – 3b)[a2 + 5ab + 9b2]
14. Factorise : 8a3 – b3 – 4ax + 2bx
Answer
We know that,
a3 – b3 = (a – b)(a2 + ab + b2) …..(1)
8a3 – b3 – 4ax + 2bx
= [(2a)3 – (b)3] – 2×(2a – b)
= (2a – b)[ (2a)2 + 2a×b + (b)2] – 2×(2a – b) [From(1)]
= (2a – b)[4a2 + 2ab + b2] – 2×(2a – b)
= (2a – b)[4a2 + 2ab + b2 – 2x]
15. Factorise : a – b – a3 + b3
Answer
we know that,
a3 – b3 = (a – b)(a2 + ab + b2) ….(1)
a – b – a3 + b3
= a – b – (a3 – b3)
= (a – b) – (a – b) [a2 + ab + b2] [From (1)]
= (a – b) [1 – a2 – ab – b2]
16. Factorise : 2x3 + 54y3 – 4x – 12y
Answer
2x3 + 54y3 – 4x – 12y
= 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)
17. Factorise : 1029 – 3x3
Answer
1029 – 3x3
= 3(343 – x3)
= 3(73 – x3)
= 3(7 – x)(72 + 7x + x2)
= 3(7 – x)(49 + 7x + x2)
18.1. Show that : 133 – 53 is divisible by 8
Answer
(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.
18.2. Show that : 353 + 273 is divisible by 62
Answer
(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.
19. Evaluate : (5.67×5.67×5.67 + 4.33×4.33×4.33)/(5.67×5.67 − 5.67×4.33 + 4.33×4.33)
Answer
Let a = 5.67 and b = 4.33
Then,
Exercise 5(E)
1. Factorise: x2 + 1/4x2 + 1 – 7x −7/2x
Answer
2. Factorise : 9a2 + 1/9a2 – 2 – 12a + 4/3a
Answer
3. Factorise : x2 + (a2+1)x/a + 1
Answer
4. Factorise : x4 + y4 – 27x2y2
Answer
5. Factorise : 4x4 + 9y4 + 11x2y2
Answer
4x4 + 9y4 + 11x2y2
= (2x2)2 + (3y2)2 + 12x2y2 – x2y2
= (2x2 + 3y2)2 – x2y2
= (2x2 + 3y2)2 – (xy)2
= ( 2x2 + 3y2 – xy )(2x2 + 3y2 + xy) [∵ a2 – b2 = (a + b)(a – b)]
6. Factorise : x2 + 1/x2 −3
Answer
7. Factorise : a – b – 4a2 + 4b2
Answer
a – b – 4a2 + 4b2
= (a – b) – 4(a2 – b2)
= (a – b ) – 4(a – b)(a + b) [∵ a2 – b2 = (a + b)(a – b)]
= (a – b)[1 – 4(a + b)]
= (a – b)[1 – 4a – 4b]
8. Factorise : (2a – 3)2 – 2 (2a – 3) (a – 1) + (a – 1)2
Answer
(2a – 3)2 – 2 (2a – 3) (a – 1) + (a – 1)2
= [(2a – 3) – (a – 1)]2
= [2a – 3 – a + 1]2
= (a – 2)2
9. Factorise : (a2 – 3a) (a2 + 3a + 7) + 10
Answer
(a2 – 3a) (a2 + 3a + 7) + 10
Let us assume , a2 – 3a = x
Then the given expression is,
(a2 – 3a)(a2 – 3a + 7) + 10
= x(x + 7) + 10
= x2 + 7x + 10
= x2 + 5x + 2x + 10
= x(x + 5) + 2 (x + 5)
= (x + 5)(x + 2)
= (a2 – 3a + 5)(a2 – 3a + 2) [resubstituting the value of x]
= (a2 – 3a + 5)(a2 – 2a – a + 2)
= (a2 – 3a + 5)(a(a – 2) – 1(a – 2))
= (a2 – 3a + 5)[(a – 1)(a – 2)]
Answer
Let us assume, a2 – a = x
Then the given expression is,
(a2 – a) (4a2 – 4a – 5) – 6
= x(4x – 5) – 6
= 4x2 – 5x – 6
= 4x2 – 8x + 3x – 6
= 4x(x – 2) + 3(x – 2)
= (4x + 3)(x – 2)
= [4( a2 – a) + 3](a2 – a – 2) [resubstitute the value of x]
= [4a2 – 4a + 3](a2 – a – 2)
= [4a2 – 4a + 3](a2 – 2a + a – 2)
= [4a2 – 4a + 3][a(a – 2) + 1(a – 2)]
= [4a2 – 4a + 3](a – 2)(a + 1)
Answer
x4 + y4 – 3x2y2
= x4 + y4 – 2x2y2 – x2y2
= (x2)2 + (y2)2 – 2x2y2 – x2y2
= (x2 – y2)2 – (xy)2
= (x2 – y2 – xy)(x2 – y2 + xy) [∵ a2 – b2 = (a + b)(a – b)]
Answer
5a2 – b2 – 4ab + 7a – 7b
= 4a2 + a2 – b2 – 4ab + 7a – 7b
= a2 – b2 + 4a2 – 4ab + 7a – 7b
= (a2 – b2) + 4a(a – b) + 7(a – b)
= (a – b)(a + b) + 4a(a – b) + 7(a – b) [∵ a2 – b2 = (a + b)(a – b)]
= (a – b)[(a + b) + 4a + 7]
= (a – b)[(a + b) + 4a + 7]
= (a – b)[5a + b + 7]
Answer
12(3x – 2y)2 – 3x + 2y – 1 = 12(3x – 2y)2 – (3x – 2y) – 1
Let us assume that 3x – 2y = a
Then the given expression is
12(3x – 2y)2 – 3x + 2y – 1
= 12a2 – 3a – 1
= 12a2 – 4a + 3a – 1
= 4a(3a – 1) + 1(3a – 1)
= ( 4a + 1 )( 3a – 1 )
= [ 4(3x – 2y) + 1][3(3x – 2y) – 1] [resubstitute the value of a]
= (12x – 8y + 1)(9x – 6y – 1)
Answer
4(2x – 3y)2 – 8x+12y – 3
= 4(2x – 3y)2 – 4(2x – 3y) – 3
Let us assume that 2x – 3y = a
Then the given expression is
4(2x – 3y)2 – 8x+12y – 3
= 4a2 – 4a – 3
= 4a2 – 6a + 2a – 3
= 2a(2a – 3) + 1(2a – 3)
= (2a – 3)(2a + 1)
= [2(2x – 3y) – 3 ][2(2x – 3y) + 1]
= (4x – 6y – 3)(4x – 6y + 1)
Answer
3 – 5x + 5y – 12(x – y)2 = 3 – 5(x – y) – 12(x – y)2
Let us assume that x – y = a
Then the given expression is
3 – 5x + 5y – 12(x – y)2
= 3 – 5a – 12a2
= 3 – 9a + 4a – 12a2
= 3(1 – 3a) + 4a(1 – 3a)
= (3 + 4a)(1 – 3a) [resubstitute the value of a]
= [3 + 4(x – y)][1 – 3(x – y)]
= (3 + 4x – 4y)(1 – 3x + 3y)
Answer
9x 2 + 3x – 8y – 64y2
= 9x2 – 64y2 + 3x – 8y
= [(3x)2 – (8y)2] + (3x – 8y)
= [(3x + 8y)(3x – 8y)] + (3x – 8y)
= (3x – 8y)(3x + 8y + 1)
Answer
2√3x2 + x – 5√3
= 2√3x2 + 6x – 5x – 5√3
= 2√3x(x + √3) – 5(x + √3)
= (2√3x – 5)(x + √3)
Answer
Answer
2(ab + cd) – a2 – b2 + c2 + d2
= 2ab + 2cd – a2 – b2 + c2 + d2
= c2 + d2 + 2cd – a2 – b2 + 2ab
= (c2 + d2 + 2cd) – (a2 + b2 – 2ab)
= (c + d)2 – (a – b)2
= (c + d + a – b)(c + d – a + b)
Answer
(987)2 – (13)2
= (987 + 13)(987 – 13)
= 1000 ×974
= 974000
Answer
(67.8)2 – (32.2)2
= (67.8 + 32.2)(67.8 – 32.2)
= 100×35.6
= 3560
Answer
Answer
