Selina Concise Solutions for Chapter 13 Factorisation Class 8 ICSE Mathematics
Exercise 13A
Factorise:
1. 15x + 5
Solution
15x + 5 = 5(3x + 1)
2. a3 – a2 + a
Solution
a3 – a2 + a
= a(a2 – a + 1)
3. 3x2 + 6x3
Solution
3x2+ 6x3
= 3x2(1 + 2x)
4. 4a2 – 8ab
Solution
4a2 – 8ab
= 4a(a – 2b)
5. 2x3b2 – 4x5b4
Solution
2x3b2 – 4x5b4
= 2x3b2(1 – 2x2b2)
6. 15x4y3 – 20x3y
Solution
Solution
15x + 5 = 5(3x + 1)
2. a3 – a2 + a
Solution
a3 – a2 + a
= a(a2 – a + 1)
3. 3x2 + 6x3
Solution
3x2+ 6x3
= 3x2(1 + 2x)
4. 4a2 – 8ab
Solution
4a2 – 8ab
= 4a(a – 2b)
5. 2x3b2 – 4x5b4
Solution
2x3b2 – 4x5b4
= 2x3b2(1 – 2x2b2)
6. 15x4y3 – 20x3y
Solution
15x4y3 – 20x3y
= 5x3y(3xy2 – 4)
7. a3b – a2b2 – b3
Solution
a3b – a2b2 – b3
= b(a3 – a2b – b2)
8. 6x2y + 9xy2 + 4y3
Solution
6x2y + 9xy2 + 4y3
= y(6x2 + 9xy + 4y2)
9. 17a6b8 – 34a4b6 + 51a2b4
Solution
17a6b8 – 34a4b6 + 51a2b4
= 17a2b4 (a4b4 – 2a2b2 + 3)
10. 3x5y – 27x4y2 + 12x3y3
Solution
3x5y – 27x4y2 + 12x3y3
= 3x3y(x2 – 9xy + 4y2)
11. x2(a – b) – y2(a – b) + z2 (a – b)
Solution
x2(a – b) - y2(a – b) + z2(a – b)
= (a – b)(x2 – y2 + z2)
12. (x + y)(a + b) + (x – y)(a + b)
Solution
(x + y)(a + b) + (x – y)(a + b)
= (a + b)(x + y + x – y)
= (a + b)(2x)
= 2x(a + b)
13. 2b(2a + b) – 3c(2a + b)
Solution
2b(2a + b) – 3c(2a + b)
= (2a + b)(2b – 3c)
14. 12abc – 6a2b2c2 + 3a3b3c3
Solution
12abc - 6a2b2c2 + 3a3b3c3
= 3abc(4 – 2abc + a2b2c2)
15. 4x(3x – 2y) – 2y(3x – 2y)
Solution
4x(3x – 2y) – 2y(x – 2y)
= (3x – 2y) (4x – 2y)
= (3x – 2y) × 2(2x – y)
= 2(3x – 2y)(2x – y)
16. (a + 2b)(3a + b) – (a + b)(a + 2b) + (a + 2b)2
Solution
(a + 2b)(3a + b) – (a + b) (a + 2b) + (a + 2b)2
= (a + 2b)(3a + b – a – b + a + 2b)
= (a + 2b)(3a + 2b)
17. 6xy(a2+ b2) + 8yz(a2 + b2) - 10xz(a2 + b2)
Solution
6xy(a2 + b2) + 8yz(a2 + b2) – 10xz(a2 + b2)
H.C.F. of 6, 8, 10 = 2, then
2(a2 + b2)(3xy + 4yz – 5xz)
= 5x3y(3xy2 – 4)
7. a3b – a2b2 – b3
Solution
a3b – a2b2 – b3
= b(a3 – a2b – b2)
8. 6x2y + 9xy2 + 4y3
Solution
6x2y + 9xy2 + 4y3
= y(6x2 + 9xy + 4y2)
9. 17a6b8 – 34a4b6 + 51a2b4
Solution
17a6b8 – 34a4b6 + 51a2b4
= 17a2b4 (a4b4 – 2a2b2 + 3)
10. 3x5y – 27x4y2 + 12x3y3
Solution
3x5y – 27x4y2 + 12x3y3
= 3x3y(x2 – 9xy + 4y2)
11. x2(a – b) – y2(a – b) + z2 (a – b)
Solution
x2(a – b) - y2(a – b) + z2(a – b)
= (a – b)(x2 – y2 + z2)
12. (x + y)(a + b) + (x – y)(a + b)
Solution
(x + y)(a + b) + (x – y)(a + b)
= (a + b)(x + y + x – y)
= (a + b)(2x)
= 2x(a + b)
13. 2b(2a + b) – 3c(2a + b)
Solution
2b(2a + b) – 3c(2a + b)
= (2a + b)(2b – 3c)
14. 12abc – 6a2b2c2 + 3a3b3c3
Solution
12abc - 6a2b2c2 + 3a3b3c3
= 3abc(4 – 2abc + a2b2c2)
15. 4x(3x – 2y) – 2y(3x – 2y)
Solution
4x(3x – 2y) – 2y(x – 2y)
= (3x – 2y) (4x – 2y)
= (3x – 2y) × 2(2x – y)
= 2(3x – 2y)(2x – y)
16. (a + 2b)(3a + b) – (a + b)(a + 2b) + (a + 2b)2
Solution
(a + 2b)(3a + b) – (a + b) (a + 2b) + (a + 2b)2
= (a + 2b)(3a + b – a – b + a + 2b)
= (a + 2b)(3a + 2b)
17. 6xy(a2+ b2) + 8yz(a2 + b2) - 10xz(a2 + b2)
Solution
6xy(a2 + b2) + 8yz(a2 + b2) – 10xz(a2 + b2)
H.C.F. of 6, 8, 10 = 2, then
2(a2 + b2)(3xy + 4yz – 5xz)
Exercise 13 B
1. Factorise : a2 + ax + ab + bx
Solution
a2 + ax + ab + bx
= (a2 + ax) + (ab + bx)
= a(a + x) + b(a + x)
= (a + x)(a + b)
2. Factorise: a2 – ab – ca + bc
Solution
a2 – ab – ca + bc
= a(a – b) – c(a – b)
= (a – b)(a – c)
3. Factorise : ab – 2b + a2 – 2a
Solution
ab – 2b + a2 – 2a
= b(a – 2) + a(a – 2)
= (a – 2)(b + a)
4. Factorise : a3 – a2 + a – 1
Solution
a3 – a2 + a – 1
= a2(a – 1) + 1(a – 1)
= (a – 1)(a2 + 1)
5. Factorise: 2a – 4b – xa + 2bx
Solution
2a – 4b – xa + 2bx
= 2(a – 2b) – x(a – 2b)
= (a – 2b)(2 – x)
6. Factorise: xy – ay – ax + a2 + bx – ab
Solution
xy – ay – ax + a2 + bx – ab
= y(x – a) – a(x – a) + b(x – a)
= (x – a)(y – a + b)
7. Factorise : 3x5 – 6x4 – 2x3 + 4x2 + x – 2
Solution
3x5 – 6x4 – 2x3 + 4x2 + x – 2
= 3x4(x – 2) – 2x2(x – 2) + 1(x – 2)
= (x – 2)(3x4 – 2x2 + 1)
8. Factorise : -x2y – x + 3xy + 3
Solution
-x2y – x + 3xy + 3
= 3 – x + 3xy – x2y (By grouping)
= 1(3 – x) + xy(3 – x)
= (3 – x)(1 + xy)
= (xy + 1)(3 – x)
9. Factorise : 6a2 – 3a2b – bc2 + 2c2
Solution
6a2 – 3a2b – bc2 + 2c2
= 6a2 – 3a2b + 2c2 – bc2 (By grouping)
= 3a2(2 – b) + c2(2 – b)
= (2 – b)(3a2 + c2)
10. Factorise : 3a2b – 12a2 – 9b + 36
Solution
3a2b – 12a2 – 9b + 36
= 3a2(b – 4) – 9(b – 4) (By grouping)
= (b – 4)(3a2 – 9)
= (b – 4)(a2 – 3)
= 3(b – 4)(a2 – 3)
11. Factorise : x2 – (a – 3)x – 3a
Solution
x2 – (a – 3)x – 3a
= x2 – ax + 3x – 3a
= x(x – a) + 3(x – a)
= (x – a)(x + 3)
12. Factorise : x2 – (b – 2)x – 2b
Solution
x2 – (b – 2)x – 2b
= x2 – bx + 2x – 2b
= x(x – b) + 2(x – b)
= (x – b) (x + 2)
13. Factorise : a(b – c) – d(c – b)
Solution
a(b – c) – d(c – b)
= a(b – c) + d(b – c)
= (b – c)(a + d)
14. Factorise : ab2 – (a – c)b – c
Solution
ab2 – (a – c)b – c
= ab2 – ab + bc – c
= ab(b – 1) + c(b – 1)
= (b – 1)(ab + c)
15. Factorise : (a2 – b2)c + (b2 – c2)a
Solution
(a2 – b2)c + (b2 – c2)a
= a2c – b2c + ab2 – ac2
= a2c – ac2 + ab2 – b2c
= ac(a – c) + b2(a – c)
= (a – c)(ac + b2)
16. a3 – a2 – ab + a + b – 1
Solution
a3 – a2 – ab + a + b – 1
= a3 – a2 – ab + b + a – 1
= a2(a – 1) – b(a – 1) + 1(a – 1)
= (a – 1)(a2 – b + 1)
17. Factorise : ab(c2+ d2) – a2cd – b2cd
Solution
ab(c2 + d2) – a2cd – b2cd
= abc2 + abd2 – a2cd – b2cd
= abc2 – a2cd – b2cd + abd2
= ac(bc – ad) – bd(bc – ad)
= (bc – ad)(ac – bd)
18. Factorise : 2ab2 – aby + 2cby – cy2
Solution
2ab2 – 2cby – aby – cy2
= 2b(ab + cy) – y(ab + cy)
= (ab + cy)(2b – y)
19. Factorise: ax + 2bx + 3cx – 3a – 6b – 9c
Solution
ax + 2bx + 3cx – 3a – 6b – 9c
= x(a + 2b + 3c) – 3(a + 2b + 3c) (By grouping)
= (a + 2b + 3c)(x – 3)
20. Factorise : 2ab2c – 2a + 3b3c – 3b – 4b2c2 + 4c
Solution
2ab2c – 2a + 3ba3c – 3b – 4b2c2 + 4c
= 2a(b2c – 1) + 3b(b2c – 1) – 4c(b2c – 1)
= (b2c – 1)(2a + 3b – 4c)
Solution
a2 + ax + ab + bx
= (a2 + ax) + (ab + bx)
= a(a + x) + b(a + x)
= (a + x)(a + b)
2. Factorise: a2 – ab – ca + bc
Solution
a2 – ab – ca + bc
= a(a – b) – c(a – b)
= (a – b)(a – c)
3. Factorise : ab – 2b + a2 – 2a
Solution
ab – 2b + a2 – 2a
= b(a – 2) + a(a – 2)
= (a – 2)(b + a)
4. Factorise : a3 – a2 + a – 1
Solution
a3 – a2 + a – 1
= a2(a – 1) + 1(a – 1)
= (a – 1)(a2 + 1)
5. Factorise: 2a – 4b – xa + 2bx
Solution
2a – 4b – xa + 2bx
= 2(a – 2b) – x(a – 2b)
= (a – 2b)(2 – x)
6. Factorise: xy – ay – ax + a2 + bx – ab
Solution
xy – ay – ax + a2 + bx – ab
= y(x – a) – a(x – a) + b(x – a)
= (x – a)(y – a + b)
7. Factorise : 3x5 – 6x4 – 2x3 + 4x2 + x – 2
Solution
3x5 – 6x4 – 2x3 + 4x2 + x – 2
= 3x4(x – 2) – 2x2(x – 2) + 1(x – 2)
= (x – 2)(3x4 – 2x2 + 1)
8. Factorise : -x2y – x + 3xy + 3
Solution
-x2y – x + 3xy + 3
= 3 – x + 3xy – x2y (By grouping)
= 1(3 – x) + xy(3 – x)
= (3 – x)(1 + xy)
= (xy + 1)(3 – x)
9. Factorise : 6a2 – 3a2b – bc2 + 2c2
Solution
6a2 – 3a2b – bc2 + 2c2
= 6a2 – 3a2b + 2c2 – bc2 (By grouping)
= 3a2(2 – b) + c2(2 – b)
= (2 – b)(3a2 + c2)
10. Factorise : 3a2b – 12a2 – 9b + 36
Solution
3a2b – 12a2 – 9b + 36
= 3a2(b – 4) – 9(b – 4) (By grouping)
= (b – 4)(3a2 – 9)
= (b – 4)(a2 – 3)
= 3(b – 4)(a2 – 3)
11. Factorise : x2 – (a – 3)x – 3a
Solution
x2 – (a – 3)x – 3a
= x2 – ax + 3x – 3a
= x(x – a) + 3(x – a)
= (x – a)(x + 3)
12. Factorise : x2 – (b – 2)x – 2b
Solution
x2 – (b – 2)x – 2b
= x2 – bx + 2x – 2b
= x(x – b) + 2(x – b)
= (x – b) (x + 2)
13. Factorise : a(b – c) – d(c – b)
Solution
a(b – c) – d(c – b)
= a(b – c) + d(b – c)
= (b – c)(a + d)
14. Factorise : ab2 – (a – c)b – c
Solution
ab2 – (a – c)b – c
= ab2 – ab + bc – c
= ab(b – 1) + c(b – 1)
= (b – 1)(ab + c)
15. Factorise : (a2 – b2)c + (b2 – c2)a
Solution
(a2 – b2)c + (b2 – c2)a
= a2c – b2c + ab2 – ac2
= a2c – ac2 + ab2 – b2c
= ac(a – c) + b2(a – c)
= (a – c)(ac + b2)
16. a3 – a2 – ab + a + b – 1
Solution
a3 – a2 – ab + a + b – 1
= a3 – a2 – ab + b + a – 1
= a2(a – 1) – b(a – 1) + 1(a – 1)
= (a – 1)(a2 – b + 1)
17. Factorise : ab(c2+ d2) – a2cd – b2cd
Solution
ab(c2 + d2) – a2cd – b2cd
= abc2 + abd2 – a2cd – b2cd
= abc2 – a2cd – b2cd + abd2
= ac(bc – ad) – bd(bc – ad)
= (bc – ad)(ac – bd)
18. Factorise : 2ab2 – aby + 2cby – cy2
Solution
2ab2 – 2cby – aby – cy2
= 2b(ab + cy) – y(ab + cy)
= (ab + cy)(2b – y)
19. Factorise: ax + 2bx + 3cx – 3a – 6b – 9c
Solution
ax + 2bx + 3cx – 3a – 6b – 9c
= x(a + 2b + 3c) – 3(a + 2b + 3c) (By grouping)
= (a + 2b + 3c)(x – 3)
20. Factorise : 2ab2c – 2a + 3b3c – 3b – 4b2c2 + 4c
Solution
2ab2c – 2a + 3ba3c – 3b – 4b2c2 + 4c
= 2a(b2c – 1) + 3b(b2c – 1) – 4c(b2c – 1)
= (b2c – 1)(2a + 3b – 4c)
Exercise 13 C
Note: a2 – b2 = (a + b)(a - b)
1. Factorise: 16 – 9x2
Solution
16 – 9x2
= (4)2 – (3x)2
= (4 + 3x)(4 – 3x)
2. Factorise : 1 – 100a2
Solution
1 – 100a2
= (1)2 – (10a)2
= (1 + 10a)(1 – 10a)
3. Factorise : 4x2 – 81y2
Solution
1. Factorise: 16 – 9x2
Solution
16 – 9x2
= (4)2 – (3x)2
= (4 + 3x)(4 – 3x)
2. Factorise : 1 – 100a2
Solution
1 – 100a2
= (1)2 – (10a)2
= (1 + 10a)(1 – 10a)
3. Factorise : 4x2 – 81y2
Solution
4x2 – 81y2
= (2x)2 – (9y)2
= (2x + 9y)(2x – 9y)
4. Factorise : 4/25 – 25b2
Solution
4/25 – 25b2
= (2/5)2 – (5b)2
= (2/5 + 5b)(2/5 – 5b)
5. Factorise : (a + 2b)2 – a2
Solution
(a + 2b)2 – a2
= (a + 2b)2 – (a)2
= (a + 2b + a)(a + 2b – a)
= (2a + 2b)(2b)
= 2(a + b)(2b)
= 2 × 2b(a + b)
= 4b(a + b)
6. Factorise : (5a – 3b)2 – 16b2
Solution
(5a – 3b)2 – 16b2
= (5a – 3b)2 – (4b2)
= (5a – 3b + 4b)(5a – 3b – 4b)
= (5a + b)(5a – 7b)
7. Factorise : a4 – (a2 – 3b2)2
Solution
a4 – (a2 – 3b2)2
= (a2)2 – (a2 – 3b2)2
= (a2 + a2 – 3b2)(a2 – a2 + 3b2)
= (2a2 – 3b2)(3b)2
= 3b2 (2a2 – 3b2)
8. Factorise : (5a – 2b)2 – (2a – b)2
Solution
(5a – 2b)2 – (2a – b)2
= (5a – 2b + 2a – b)(5a – 2b – 2a + b)
= (7a – 3b)(3a – b)
9. Factorise : 1 – 25(a + b)2
Solution
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)
10. Factorise : 4(2a + b)2 – (a – b)2
Solution
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)
11. Factorise : 25(2x + y)2 – 16(x – y)2
Solution
25(2x + y)2 – 16(x – y)2
= [5(2x + y)]2 – [4(x – y)]2
= (10x + 5y)2 – (4x – 4y)2
= (10x + 5y + 4x – 4y)(10x + 5y – 4x + 4y)
= (14x + y)(6x + 9y)
= (14x + y) 3(2x + 3y)
= 3(14x + y)(2x + 3y)
12. Factorise : 49(x – y)2 – 9(2x + y)2
Solution
[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)
13. Evaluate : (6 2/3)2 – (2 1/3)2
Solution
(6 2/3)2 – (2 1/3)2
= (20/3)2 – (7/3)2
= (20/3 + 7/3)(20/3 – 7/3)
= (27/3)(13/3)
= 9 × 13/3
= 39
14. Evaluate : (7 3/10)2 – (2 1/10)2
Solution
(7 3/10)2 – (2 1/10)2
= (73/10)2 – (21/10)2
= (73/10 + 21/10)(73/10 – 21/10)
= (94/10)(52/10)
= (47/5)(26/5)
= 1222/25
= 48 22/25
15. Evaluate : (0.7)2 – (0.3)2
Solution
(0.7)2 – (0.3)2
= (0.7 + 0.3)(0.7 – 0.3)
= 1 × 0.4
= 0.4
16. Evaluate : (4.5)2 – (1.5)2
Solution
(4.5)2 – (1.5)2
= (4.5 + 1.5)(4.5 – 1.5)
= 6 × 3
= 18
17. Factorise : 75(x + y)2 – 48(x – y)2
Solution
75(x + y)2 – 48(x – y)2
= 3[25(x + y)2 – 16(x – y)2]
= 3[{5(x + y)2} – {4(x – y)}2]
Using a2 – b2 = (a + b)(a – b)
= 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)
18. Factorise : a2 + 4a + 4 – b2
Solution
a2 + 4a + 4 – b2
[∵ (a + b)2 = a2 + 2ab + b2 and a2 – b2 = (a + b)(a – b)]
= (2x)2 – (9y)2
= (2x + 9y)(2x – 9y)
4. Factorise : 4/25 – 25b2
Solution
4/25 – 25b2
= (2/5)2 – (5b)2
= (2/5 + 5b)(2/5 – 5b)
5. Factorise : (a + 2b)2 – a2
Solution
(a + 2b)2 – a2
= (a + 2b)2 – (a)2
= (a + 2b + a)(a + 2b – a)
= (2a + 2b)(2b)
= 2(a + b)(2b)
= 2 × 2b(a + b)
= 4b(a + b)
6. Factorise : (5a – 3b)2 – 16b2
Solution
(5a – 3b)2 – 16b2
= (5a – 3b)2 – (4b2)
= (5a – 3b + 4b)(5a – 3b – 4b)
= (5a + b)(5a – 7b)
7. Factorise : a4 – (a2 – 3b2)2
Solution
a4 – (a2 – 3b2)2
= (a2)2 – (a2 – 3b2)2
= (a2 + a2 – 3b2)(a2 – a2 + 3b2)
= (2a2 – 3b2)(3b)2
= 3b2 (2a2 – 3b2)
8. Factorise : (5a – 2b)2 – (2a – b)2
Solution
(5a – 2b)2 – (2a – b)2
= (5a – 2b + 2a – b)(5a – 2b – 2a + b)
= (7a – 3b)(3a – b)
9. Factorise : 1 – 25(a + b)2
Solution
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)
10. Factorise : 4(2a + b)2 – (a – b)2
Solution
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)
11. Factorise : 25(2x + y)2 – 16(x – y)2
Solution
25(2x + y)2 – 16(x – y)2
= [5(2x + y)]2 – [4(x – y)]2
= (10x + 5y)2 – (4x – 4y)2
= (10x + 5y + 4x – 4y)(10x + 5y – 4x + 4y)
= (14x + y)(6x + 9y)
= (14x + y) 3(2x + 3y)
= 3(14x + y)(2x + 3y)
12. Factorise : 49(x – y)2 – 9(2x + y)2
Solution
[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)
13. Evaluate : (6 2/3)2 – (2 1/3)2
Solution
(6 2/3)2 – (2 1/3)2
= (20/3)2 – (7/3)2
= (20/3 + 7/3)(20/3 – 7/3)
= (27/3)(13/3)
= 9 × 13/3
= 39
14. Evaluate : (7 3/10)2 – (2 1/10)2
Solution
(7 3/10)2 – (2 1/10)2
= (73/10)2 – (21/10)2
= (73/10 + 21/10)(73/10 – 21/10)
= (94/10)(52/10)
= (47/5)(26/5)
= 1222/25
= 48 22/25
15. Evaluate : (0.7)2 – (0.3)2
Solution
(0.7)2 – (0.3)2
= (0.7 + 0.3)(0.7 – 0.3)
= 1 × 0.4
= 0.4
16. Evaluate : (4.5)2 – (1.5)2
Solution
(4.5)2 – (1.5)2
= (4.5 + 1.5)(4.5 – 1.5)
= 6 × 3
= 18
17. Factorise : 75(x + y)2 – 48(x – y)2
Solution
75(x + y)2 – 48(x – y)2
= 3[25(x + y)2 – 16(x – y)2]
= 3[{5(x + y)2} – {4(x – y)}2]
Using a2 – b2 = (a + b)(a – b)
= 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)
18. Factorise : a2 + 4a + 4 – b2
Solution
a2 + 4a + 4 – b2
[∵ (a + b)2 = a2 + 2ab + b2 and a2 – b2 = (a + b)(a – b)]
= (a)2 + 2 × a × 2 + (2)2 – (b)2
= (a + 2)2 – (b)2
= (a + 2 + b)(a + 2 – b)
= (a + b + 2)(a – b + 2)
19. Factorise : a2 – b2 – 2b – 1
Solution
a2 - b2 – 2b – 1
= a2 – (b2 + 2b + 1)
= (a + 2)2 – (b)2
= (a + 2 + b)(a + 2 – b)
= (a + b + 2)(a – b + 2)
19. Factorise : a2 – b2 – 2b – 1
Solution
a2 - b2 – 2b – 1
= a2 – (b2 + 2b + 1)
[∵ (a + b)2 = a2 + 2ab + b2 and a2 – b2 = (a + b)(a – b)]
= (a)2 – (b + 1)2
= (a + b + 1)(a – b – 1)
20. Factorise: x2 + 6x + 9 – 4y2
Solution
x2 + 6x + 9 – 4y2
= (x)2 + 2 × x × 3 + (3)2 – (2y)2
[∵ (a + b)2 = a2 + 2ab + b2 and a2 – b2 = (a + b)(a – b)]
= (x + 3)2 – (2y)2
= (x + 3 + 2y)(x + 3 – 2y)
= (x + 2y + 3)(x – 2y + 3)
= (a)2 – (b + 1)2
= (a + b + 1)(a – b – 1)
20. Factorise: x2 + 6x + 9 – 4y2
Solution
x2 + 6x + 9 – 4y2
= (x)2 + 2 × x × 3 + (3)2 – (2y)2
[∵ (a + b)2 = a2 + 2ab + b2 and a2 – b2 = (a + b)(a – b)]
= (x + 3)2 – (2y)2
= (x + 3 + 2y)(x + 3 – 2y)
= (x + 2y + 3)(x – 2y + 3)
Exercise 13 D
1. Factorise : x2 + 6x + 8
Solution
x2 + 6x + 8
= x2 + 4x + 2x + 8
= x(x + 4) + 2(x + 4)
= (x + 4)(x + 2)
2. Factorise : x2 + 4x + 3
Solution
x2 + 4x + 3
= x2 + 3x + x + 3
= x(x + 3) + 1(x + 3)
= (x + 3)(x + 1)
3. a2 + 5a + 6
Solution
a2 + 5a + 6
= a2 + 3a + 2a + 6
= a(a + 3) + 2(a + 3)
= (a + 3)(a + 2)
4. a2 – 5a + 6
Solution
a2 – 5a + 6
= a2 – 3a – 2a + 6
= a(a – 3) - 2(a – 3)
= (a – 3)(a – 2)
5. Factorise : a2 + 5a – 6
Solution
a2 + 5a – 6
= a2 + 6a – a – 6
= a(a + 6) – 1(a + 6)
= (a + 6)(a – 1)
6. Factorise : x2 + 5xy + 4y2
Solution
x2 + 5xy + 4y2
= x2 + 4xy + xy + 4y2
= x(x + 4y) + y(x + 4y)
= (x + 4y)(x + y)
7. Factorise : a2 – 3a – 40
Solution
a2 – 3a – 40
= a2 – 8a + 5a – 40
= a(a – 8) + 5(a – 8)
= (a – 8)(a + 5)
8. Factorise : x2 – x – 72
Solution
x2 – x – 72
= x2 – 9x + 8x – 72
= x(x – 9) + 8(x – 9)
= (x – 9)(x + 8)
9. Factorise : x2 – 10xy + 24y2
Solution
x2 – 10xy + 24y2
= x2 – 6xy – 4xy + 24y2
= x(x – 6y) – 4y(x – 6y)
= (x – 6y)(x – 4y)
10. Factorise : 2a2 + 7a + 6
Solution
2a2 + 7a + 6
= 2a2 + 4a + 3a + 6
= 2a(a + 2) + 3(a + 2)
= (a + 2)(2a + 3)
11. Factorise : 3a2 – 5a + 2
Solution
3a2 – 5a + 2
= 3a2 – 3a – 2a + 2
= 3a(a – 1) – 2(a – 1)
= (a – 1)(3a – 2)
12. Factorise : 7b2 – 8b + 1
Solution
7b2 – 8b + 1
= 7b2 – 7b – b + 1
= 7b(b – 1) - 1(b – 1)
= (b - 1)(7b - 1)
13. Factorise : 2a2 – 17ab + 26b2
Solution
2a2 – 17ab + 26b2
= 2a2 – 13ab – 4ab + 26b2
= a(2a – 13b) – 2b(2a – 13b)
= (2a – 13b)(a – 2b)
14. Factorise : 2x2 + xy – 6y2
Solution
2x2 + xy – 6y2
= 2x2 + 4xy – 3xy – 6y2
= 2x(x + 2y) – 3y(x + 2y)
= (x + 2y)(2x – 3y)
15. Factorise : 4c2 + 3c – 10
Solution
4c2 + 3c – 10
= 4c2 + 8c – 5c – 10
= 4c(c + 2) – 5(c + 2)
= (c + 2)(4c – 5)
16. Factorise : 14x2 + x – 3
Solution
14x2 + x – 3
= 14x2 + 7x – 6x – 3
= 7x(2x + 1) – 3(2x + 1)
= (2x + 1)(7x – 3)
17. Factorise : 6 + 7b – 3b2
Solution
6 + 7b – 3b2
= 6 + 9b – 2b – 3b2
= 3(2 + 3b) – b(2 + 3b)
= (2 + 3b)(3 – b)
18. Factorise : 5 + 7x – 6x2
Solution
5 + 7x - 6x2
= 5 + 10x – 3x – 6x2
= 5(1 + 2x) – 3x(1 + 2x)
= (1 + 2x)(5 – 3x)
19. Factorise: 4 + y – 14y2
Solution
4 + y – 14y2
= 4 + 8y – 7y – 14y2
= 4(1 + 2y) – 7y(1 + 2y)
= (1 + 2y)(4 – 7y)
20. Factorise : 5 + 3a – 14a2
Solution
5 + 3a – 14a2
= 5 + 10a – 7a – 14a2
= 5(1 + 2a) – 7a(1 + 2a)
= (1 + 2a)(5 – 7a)
21. Factorise : (2a + b)2 + 5(2a + b) + 6
Solution
Let (2a + b) = x
(2a + b)2 = x2
(2a + b)2 + 5(2a + b) + 6
= x2 + 5x + 6
= x2 + 3x + 2x + 6
= x(x + 3) + 2(x + 3)
= (x + 3)(x + 2)
= (2a + b + 3)(2a + b + 2)
(Substituting the value of x)
22. Factorise : 1 – (2x + 3y) – 6(2x + 3y)2
Solution
Let (2x + 3y) = a
∴ (2x + 3y)2 = a2
∴ 1 – (2x + 3y) – 6(2x + 3y)2
= 1 – a – 6a2
= 1 – 3a + 2a – 6a2
= 1(1 – 3a) + 2a(1 – 3a)
= (1 – 3a) (1 + 2a)
= [1 – 3(2x + 3y)][1 + 2(2x + 3y)]
(Substituting the value of a)
= (1 – 6x – 9y)(1 + 4x + 6y)
23. Factorise : (x – 2y)2 – 12(x – 2y) + 32
Solution
Let (x – 2y) = a
∴ (x – 2y)2 = a2
∴ (x – 2y)2 - 12(x – 2y) + 32
= a2 – 12a + 32
= a2 – 8a – 4a + 32
= a(a – 8) – 4(a – 8)
= (a – 8)(a – 4)
= (x – 2y – 8)(x – 2y – 4)
(Substituting the value of a)
24. Factorise : 8 + 6(a + b) – 5(a + b)2
Solution
Let a + b = x
(a + b)2 = x2
8 + 6(a + b) – 5(a + b)2
= 8 + 6x – 5x2
= 8 + 10x - 4x - 5x2
= 2(4 + 5x) - x(4 + 5x)
= (4 + 5x)(2 – x)
= [4 + 5(a + b)] [2 – (a + b)]
(Substituting the value of x)
= [4 + 5a + 5b][2 – a – b]
25. Factorise : 2(x + 2y)2 – 5(x + 2y) + 2
Solution
2(x + 2y)2 – 5(x + 2y) + 2
Let x + 2y = a, then
2a2 – 5a + 2
⇒ 2a2 – a – 4a + 2
= a(2a – 1) – 2(2a – 1)
= (2a – 1) (a – 2)
= {2(x + 2y – 1)} {(x + 2y) – 2}
= (2x + 4y - 2)(x + 2y – 2)
{∵ 2×2 = 4, -5 = - 1 - 4, 4 = (-1) × (-4)}
Solution
x2 + 6x + 8
= x2 + 4x + 2x + 8
= x(x + 4) + 2(x + 4)
= (x + 4)(x + 2)
2. Factorise : x2 + 4x + 3
Solution
x2 + 4x + 3
= x2 + 3x + x + 3
= x(x + 3) + 1(x + 3)
= (x + 3)(x + 1)
3. a2 + 5a + 6
Solution
a2 + 5a + 6
= a2 + 3a + 2a + 6
= a(a + 3) + 2(a + 3)
= (a + 3)(a + 2)
4. a2 – 5a + 6
Solution
a2 – 5a + 6
= a2 – 3a – 2a + 6
= a(a – 3) - 2(a – 3)
= (a – 3)(a – 2)
5. Factorise : a2 + 5a – 6
Solution
a2 + 5a – 6
= a2 + 6a – a – 6
= a(a + 6) – 1(a + 6)
= (a + 6)(a – 1)
6. Factorise : x2 + 5xy + 4y2
Solution
x2 + 5xy + 4y2
= x2 + 4xy + xy + 4y2
= x(x + 4y) + y(x + 4y)
= (x + 4y)(x + y)
7. Factorise : a2 – 3a – 40
Solution
a2 – 3a – 40
= a2 – 8a + 5a – 40
= a(a – 8) + 5(a – 8)
= (a – 8)(a + 5)
8. Factorise : x2 – x – 72
Solution
x2 – x – 72
= x2 – 9x + 8x – 72
= x(x – 9) + 8(x – 9)
= (x – 9)(x + 8)
9. Factorise : x2 – 10xy + 24y2
Solution
x2 – 10xy + 24y2
= x2 – 6xy – 4xy + 24y2
= x(x – 6y) – 4y(x – 6y)
= (x – 6y)(x – 4y)
10. Factorise : 2a2 + 7a + 6
Solution
2a2 + 7a + 6
= 2a2 + 4a + 3a + 6
= 2a(a + 2) + 3(a + 2)
= (a + 2)(2a + 3)
11. Factorise : 3a2 – 5a + 2
Solution
3a2 – 5a + 2
= 3a2 – 3a – 2a + 2
= 3a(a – 1) – 2(a – 1)
= (a – 1)(3a – 2)
12. Factorise : 7b2 – 8b + 1
Solution
7b2 – 8b + 1
= 7b2 – 7b – b + 1
= 7b(b – 1) - 1(b – 1)
= (b - 1)(7b - 1)
13. Factorise : 2a2 – 17ab + 26b2
Solution
2a2 – 17ab + 26b2
= 2a2 – 13ab – 4ab + 26b2
= a(2a – 13b) – 2b(2a – 13b)
= (2a – 13b)(a – 2b)
14. Factorise : 2x2 + xy – 6y2
Solution
2x2 + xy – 6y2
= 2x2 + 4xy – 3xy – 6y2
= 2x(x + 2y) – 3y(x + 2y)
= (x + 2y)(2x – 3y)
15. Factorise : 4c2 + 3c – 10
Solution
4c2 + 3c – 10
= 4c2 + 8c – 5c – 10
= 4c(c + 2) – 5(c + 2)
= (c + 2)(4c – 5)
16. Factorise : 14x2 + x – 3
Solution
14x2 + x – 3
= 14x2 + 7x – 6x – 3
= 7x(2x + 1) – 3(2x + 1)
= (2x + 1)(7x – 3)
17. Factorise : 6 + 7b – 3b2
Solution
6 + 7b – 3b2
= 6 + 9b – 2b – 3b2
= 3(2 + 3b) – b(2 + 3b)
= (2 + 3b)(3 – b)
18. Factorise : 5 + 7x – 6x2
Solution
5 + 7x - 6x2
= 5 + 10x – 3x – 6x2
= 5(1 + 2x) – 3x(1 + 2x)
= (1 + 2x)(5 – 3x)
19. Factorise: 4 + y – 14y2
Solution
4 + y – 14y2
= 4 + 8y – 7y – 14y2
= 4(1 + 2y) – 7y(1 + 2y)
= (1 + 2y)(4 – 7y)
20. Factorise : 5 + 3a – 14a2
Solution
5 + 3a – 14a2
= 5 + 10a – 7a – 14a2
= 5(1 + 2a) – 7a(1 + 2a)
= (1 + 2a)(5 – 7a)
21. Factorise : (2a + b)2 + 5(2a + b) + 6
Solution
Let (2a + b) = x
(2a + b)2 = x2
(2a + b)2 + 5(2a + b) + 6
= x2 + 5x + 6
= x2 + 3x + 2x + 6
= x(x + 3) + 2(x + 3)
= (x + 3)(x + 2)
= (2a + b + 3)(2a + b + 2)
(Substituting the value of x)
22. Factorise : 1 – (2x + 3y) – 6(2x + 3y)2
Solution
Let (2x + 3y) = a
∴ (2x + 3y)2 = a2
∴ 1 – (2x + 3y) – 6(2x + 3y)2
= 1 – a – 6a2
= 1 – 3a + 2a – 6a2
= 1(1 – 3a) + 2a(1 – 3a)
= (1 – 3a) (1 + 2a)
= [1 – 3(2x + 3y)][1 + 2(2x + 3y)]
(Substituting the value of a)
= (1 – 6x – 9y)(1 + 4x + 6y)
23. Factorise : (x – 2y)2 – 12(x – 2y) + 32
Solution
Let (x – 2y) = a
∴ (x – 2y)2 = a2
∴ (x – 2y)2 - 12(x – 2y) + 32
= a2 – 12a + 32
= a2 – 8a – 4a + 32
= a(a – 8) – 4(a – 8)
= (a – 8)(a – 4)
= (x – 2y – 8)(x – 2y – 4)
(Substituting the value of a)
24. Factorise : 8 + 6(a + b) – 5(a + b)2
Solution
Let a + b = x
(a + b)2 = x2
8 + 6(a + b) – 5(a + b)2
= 8 + 6x – 5x2
= 8 + 10x - 4x - 5x2
= 2(4 + 5x) - x(4 + 5x)
= (4 + 5x)(2 – x)
= [4 + 5(a + b)] [2 – (a + b)]
(Substituting the value of x)
= [4 + 5a + 5b][2 – a – b]
25. Factorise : 2(x + 2y)2 – 5(x + 2y) + 2
Solution
2(x + 2y)2 – 5(x + 2y) + 2
Let x + 2y = a, then
2a2 – 5a + 2
⇒ 2a2 – a – 4a + 2
= a(2a – 1) – 2(2a – 1)
= (2a – 1) (a – 2)
= {2(x + 2y – 1)} {(x + 2y) – 2}
= (2x + 4y - 2)(x + 2y – 2)
{∵ 2×2 = 4, -5 = - 1 - 4, 4 = (-1) × (-4)}
Exercise 13 E
1. In each case find whether the trinomial is a perfect square or not:
(i) x2+ 14x + 49
(ii) a2 – 10a + 25
(iii) 4x2 + 4x + 1
(iv) 9b2 + 12b + 16
(v) 16x2 – 16xy + y2
(vi) x2 – 4x + 16
Solution
(i) x2 + 14x + 49
= (x)2 + 2 × x + 7 + (7)2
= (x + 7)2
[∵ a2 + 2ab + b2 = (a + b)2]
∴ The given trinomial x2 + 14x + 49 is a perfect square.
(ii) a2 – 10a + 25
= (a)2 – 2 × a × 5 + (5)2
= (a - 5)2
[∵ a2 - 2ab + b2 = (a + b)2]
∴ The given trinomial a2 – 10a + 25 is a perfect square.
(iii) 4x2 + 4x + 1
= (2x)2 + 2 × 2x × 1 + (1)2
= (2x + 1)2
[∵ a2 + 2ab + b2 = (a + b)2]
∴ The given trinomial 4x2 + 4x = 1 is a perfect square.
(iv) 9b2 + 12b + 16
= (3b)2 + 3b × 4 + (4)2
= x2 + xy + y2
[Taking 3b = x, and 4 = y]
∴ The given trinomial cannot be expressed as x2 + 2xy + y2. Hence, it is not a perfect square.
(v) 16x2 – 16xy + y2
= (4x)2 – 4 × 4x × y + (y)2
= a2 – 4ab + b2
[Taking 4x = a, and y = b]
∴ The given trinomial cannot be expressed as a2 – 2ab + b2.
∴ It is not a perfect square.
(vi) x2 – 4x + 16
= (x)2 – x × 4 + (4)2
= a – ab + b2
[Taking x = a, and 4 = b]
∴ The given trinomial cannot be expressed as a2 – 2ab + b2.
Hence, it is not a perfect square.
2. Factorise completely 2 – 8x2.
Solution
2 – 8x2 = 2(1 – 4x2)
= 2[(1)2 – (2x)2]
= 2(1 + 2x)(1 – 2x)
Note: a2 – b2 = (a + b)(a – b)
3. Factorise completely: 8x2y – 18y3
Solution
8x2 – 18y3
= 2y(4x2 – 9y2)
= 2y[(2x)2 – (3y)2]
= 2y(2x + 3y)(2x – 3y)
4. Factorise completely : ax2 – ay2
Solution
ax2 – ay2
= a(x2 – y2)
= a(x + y)(x – y)
5. Factorise completely : 25x3 – x
Solution
25x3 – x
= x(25x2 – 1)
= x[(5x)2 – (1)2]
= x(5x + 1) (5x – 1)
6. Factorise completely : a4 – b4
Solution
a4 – b4
= (a2)2 – (b2)2
= (a2 + b2) (a2 – b2)
= (a2 + b2) (a + b)(a – b)
7. Factorise completely : 16x4 – 81y4
Solution
16x4 – 81y4
= (4x2)2 – (9y2)2
= (4x2 + 9y2)(4x2 – 9y2)
= (4x2 + 9y2)[(2x)2 – (3y)2]
= (4x2 + 9y2)(2x + 3y)(2x – 3y)
8. Factorise completely : 625 – x4
Solution
625 – x4 = (25)2 – (x2)2
= (25 + x2)(25 – x2)
= (25 + x2)[(5)2 – (x)2]
= (25 + x2)(5 + x)(5 – x)
9. Factorise completely : x2 – y2 – 3x – 3y
Solution
x2 – y2 – 3x – 3y
= (x2 – y2) – 3(x + y)
= (x + y)(x – y) – 3(x + y)
= (x + y)(x – y – 3)
10. Factorise completely : x2 – y2 – 2x + 2y
Solution
x2 – y2 – 2x + 2y
= (x2 – y2) – 2(x – y)
= (x + y)(x – y) – 2(x – y)
= (x – y)(x + y – 2)
11. Factorise completely : 3x2 + 15x – 72
Solution
3x2 +15x – 72
= 3(x2 + 5x – 24)
= 3[x2 + 8x – 3x – 24]
= 3[x(x + 8) – 3(x + 8)
= 3[(x + 8)(x – 3)]
= 3(x + 8)(x – 3)
12. Factorise completely : 2a2 – 8a – 64
Solution
2a2 – 8a – 64
= 2[a2 – 4a – 32]
= 2[a2 – 8a + 4a – 32]
= 2[a(a – 8) + 4(a – 8)]
= 2[(a – 8)(a + 4)]
= 2(a – 8)(a + 4)
13. Factorise completely : 5b2 + 45b + 90
Solution
5b2 + 45b + 90
= 5[b2 + 9b + 18]
= 5[b2 + 6b + 3b + 18]
= 5[b(b + 6) + 3(b + 6)]
= 5[(b + 6)(b + 3)]
= 5(b + 6)(b + 3)
14. Factorise completely : 3x2y + 11xy + 6y
Solution
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)
15. Factorise completely : 5ap2 + 11ap + 2a
Solution
5ap2 + 11ap + 2a
= a[5p2 + 11p + 2]
= a[5p2 + 10p + p + 2]
= a[5p(p + 2) + 1(p + 2)]
= a[(p + 2)(5p + 1)]
= a(p + 2)(5p + 1)
16. Factorise completely : a2 + 2ab + b2 – c2
Solution
a2 + 2ab + b2 – c2
= (a2 + 2ab + b2) – c2
= (a + b)2 – (c2)
= (a + b + c)(a + b – c)
17. Factorise completely : x2 + 6xy + 9y2 + x + 3y
Solution
x2 + 6xy + 9y2 + x + 3y
= [(x)2 + 2 × x × 3y + (3y)2] + (x + 3y)
= [x + 3y]2 + (x + 3y)
= (x + 3y)(x + 3y) + (x + 3y)
= (x + 3y)(x + 3y + 1)
18. Factorise completely : 4a2 – 12ab + 9b2 + 4a – 6b
Solution
[4a2 – 12ab + 9b2] + (4a – 6b)
= [(2a)2 – 2 × 2a × 3b + (3b)2] + 2(2a – 3b)
= (2a – 3b)2 + 2(2a – 3b)
= (2a – 3b)(2a – 3b + 2)
19. Factorise completely : 2a2b2 – 98b4
Solution
2a2b2 – 98b4
= 2b2(a2 – 49b2)
= 2b2[(a)2 – (7b)2]
= 2b2(a + 7b)(a – 7b)
20. Factorise completely : a2 – 16b2 – 2a – 8b
Solution
(a2 – 16b)2 – 2a – 8b
= [(a)4 – (4b)2] - 2(a + 4b)
= (a + 4b)(a – 4b) – 2(a + 4b)
= (a + 4b)(a – 4b – 2)
(i) x2+ 14x + 49
(ii) a2 – 10a + 25
(iii) 4x2 + 4x + 1
(iv) 9b2 + 12b + 16
(v) 16x2 – 16xy + y2
(vi) x2 – 4x + 16
Solution
(i) x2 + 14x + 49
= (x)2 + 2 × x + 7 + (7)2
= (x + 7)2
[∵ a2 + 2ab + b2 = (a + b)2]
∴ The given trinomial x2 + 14x + 49 is a perfect square.
(ii) a2 – 10a + 25
= (a)2 – 2 × a × 5 + (5)2
= (a - 5)2
[∵ a2 - 2ab + b2 = (a + b)2]
∴ The given trinomial a2 – 10a + 25 is a perfect square.
(iii) 4x2 + 4x + 1
= (2x)2 + 2 × 2x × 1 + (1)2
= (2x + 1)2
[∵ a2 + 2ab + b2 = (a + b)2]
∴ The given trinomial 4x2 + 4x = 1 is a perfect square.
(iv) 9b2 + 12b + 16
= (3b)2 + 3b × 4 + (4)2
= x2 + xy + y2
[Taking 3b = x, and 4 = y]
∴ The given trinomial cannot be expressed as x2 + 2xy + y2. Hence, it is not a perfect square.
(v) 16x2 – 16xy + y2
= (4x)2 – 4 × 4x × y + (y)2
= a2 – 4ab + b2
[Taking 4x = a, and y = b]
∴ The given trinomial cannot be expressed as a2 – 2ab + b2.
∴ It is not a perfect square.
(vi) x2 – 4x + 16
= (x)2 – x × 4 + (4)2
= a – ab + b2
[Taking x = a, and 4 = b]
∴ The given trinomial cannot be expressed as a2 – 2ab + b2.
Hence, it is not a perfect square.
2. Factorise completely 2 – 8x2.
Solution
2 – 8x2 = 2(1 – 4x2)
= 2[(1)2 – (2x)2]
= 2(1 + 2x)(1 – 2x)
Note: a2 – b2 = (a + b)(a – b)
3. Factorise completely: 8x2y – 18y3
Solution
8x2 – 18y3
= 2y(4x2 – 9y2)
= 2y[(2x)2 – (3y)2]
= 2y(2x + 3y)(2x – 3y)
4. Factorise completely : ax2 – ay2
Solution
ax2 – ay2
= a(x2 – y2)
= a(x + y)(x – y)
5. Factorise completely : 25x3 – x
Solution
25x3 – x
= x(25x2 – 1)
= x[(5x)2 – (1)2]
= x(5x + 1) (5x – 1)
6. Factorise completely : a4 – b4
Solution
a4 – b4
= (a2)2 – (b2)2
= (a2 + b2) (a2 – b2)
= (a2 + b2) (a + b)(a – b)
7. Factorise completely : 16x4 – 81y4
Solution
16x4 – 81y4
= (4x2)2 – (9y2)2
= (4x2 + 9y2)(4x2 – 9y2)
= (4x2 + 9y2)[(2x)2 – (3y)2]
= (4x2 + 9y2)(2x + 3y)(2x – 3y)
8. Factorise completely : 625 – x4
Solution
625 – x4 = (25)2 – (x2)2
= (25 + x2)(25 – x2)
= (25 + x2)[(5)2 – (x)2]
= (25 + x2)(5 + x)(5 – x)
9. Factorise completely : x2 – y2 – 3x – 3y
Solution
x2 – y2 – 3x – 3y
= (x2 – y2) – 3(x + y)
= (x + y)(x – y) – 3(x + y)
= (x + y)(x – y – 3)
10. Factorise completely : x2 – y2 – 2x + 2y
Solution
x2 – y2 – 2x + 2y
= (x2 – y2) – 2(x – y)
= (x + y)(x – y) – 2(x – y)
= (x – y)(x + y – 2)
11. Factorise completely : 3x2 + 15x – 72
Solution
3x2 +15x – 72
= 3(x2 + 5x – 24)
= 3[x2 + 8x – 3x – 24]
= 3[x(x + 8) – 3(x + 8)
= 3[(x + 8)(x – 3)]
= 3(x + 8)(x – 3)
12. Factorise completely : 2a2 – 8a – 64
Solution
2a2 – 8a – 64
= 2[a2 – 4a – 32]
= 2[a2 – 8a + 4a – 32]
= 2[a(a – 8) + 4(a – 8)]
= 2[(a – 8)(a + 4)]
= 2(a – 8)(a + 4)
13. Factorise completely : 5b2 + 45b + 90
Solution
5b2 + 45b + 90
= 5[b2 + 9b + 18]
= 5[b2 + 6b + 3b + 18]
= 5[b(b + 6) + 3(b + 6)]
= 5[(b + 6)(b + 3)]
= 5(b + 6)(b + 3)
14. Factorise completely : 3x2y + 11xy + 6y
Solution
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)
15. Factorise completely : 5ap2 + 11ap + 2a
Solution
5ap2 + 11ap + 2a
= a[5p2 + 11p + 2]
= a[5p2 + 10p + p + 2]
= a[5p(p + 2) + 1(p + 2)]
= a[(p + 2)(5p + 1)]
= a(p + 2)(5p + 1)
16. Factorise completely : a2 + 2ab + b2 – c2
Solution
a2 + 2ab + b2 – c2
= (a2 + 2ab + b2) – c2
= (a + b)2 – (c2)
= (a + b + c)(a + b – c)
17. Factorise completely : x2 + 6xy + 9y2 + x + 3y
Solution
x2 + 6xy + 9y2 + x + 3y
= [(x)2 + 2 × x × 3y + (3y)2] + (x + 3y)
= [x + 3y]2 + (x + 3y)
= (x + 3y)(x + 3y) + (x + 3y)
= (x + 3y)(x + 3y + 1)
18. Factorise completely : 4a2 – 12ab + 9b2 + 4a – 6b
Solution
[4a2 – 12ab + 9b2] + (4a – 6b)
= [(2a)2 – 2 × 2a × 3b + (3b)2] + 2(2a – 3b)
= (2a – 3b)2 + 2(2a – 3b)
= (2a – 3b)(2a – 3b + 2)
19. Factorise completely : 2a2b2 – 98b4
Solution
2a2b2 – 98b4
= 2b2(a2 – 49b2)
= 2b2[(a)2 – (7b)2]
= 2b2(a + 7b)(a – 7b)
20. Factorise completely : a2 – 16b2 – 2a – 8b
Solution
(a2 – 16b)2 – 2a – 8b
= [(a)4 – (4b)2] - 2(a + 4b)
= (a + 4b)(a – 4b) – 2(a + 4b)
= (a + 4b)(a – 4b – 2)
Exercise 13 F
1. Factorise :
(i) 6x3 – 8x2
(ii) 35a3b2c + 42ab2c2
(iii) 36x2y2 – 30x3y3 + 48x3y2
(iv) 8(2a + 3b)3 – 12(2a + 3b)2
(v) 9a(x – 2y)4 – 12a(x – 2y)3
Solution
(i) 6x3 – 8x2 = 2x2(3x – 4)
(ii) 35a3b3c + 42ab2c2
= 7ab2c (5a2 + 6c)
(iii) 36x2y2 – 30x3y3 + 48x3y3
= 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 (3x – 6y – 4)
2. Factorise :
(i) a2 – ab – 3a + 3b
(ii) x2y – xy2 + 5x – 5y
(iii) a2 – ab(1 – b) – b3
(iv) xy2 + (x – 1)y – 1
(v) (ax + by)2 + (bx – ay)2
(vi) ab(x2 + y2) – xy(a2 + b2)
(vii) m – 1 – (m – 1)2 + am – a
Solution
(i) a2 – ab – 3a + 3b
= a(a - b) – 3(a – b)
= (a – b)(a – 3)
(ii) x2y – xy2 + 5x – 5y
= xy(x – y) + 5(x – y)
= (x – y)(xy + 5)
(iii) a2 – ab(1 – ab) – b3
= a2 – ab + ab2 – b3
= a(a – b) + b2(a – b)
= (a – b)(a + b)2
(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)2
= a2x2 + b2y2 + 2abxy + b2x2 + a2y2 – 2abxy
= a2x2 + b2y2 + b2x2 + ay2
= a2x2 + a2y2 + b2x2 + b2y2
= a2(x2 + y2) + b2(x2 + y2)
= (x2 + y2)(a2 + b2)
(vi) ab(x2 + y2) – xy (a2 + b2)
= abx2 + aby2 – a2xy – b2xy
= abx2 – a2xy + aby2 – 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)(1 – m + 1 + a)
= (m – 1)(2 – m + a)
3. Factorise :
(i) a2 – (b – c)2
(ii) 25(2x – y)2 – 16(x – 2y)2
(iii) 16(5x + 4)2 – 9(3x – 2)2
(iv) 9x2 – 1/16
(v) 25(x – 2y)2 – 4
Solution
(i) a2 – (b – c)2
= (a – (b – c))(a + b – c)
[a2 – b2 = (a – b)(a + b)]
= (a – b + c)(a + b – c)
(ii) 25(2x – y)2 – 16(x – 2y)2
= (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]
[a2 – b2 = (a – b)(a + b)]
= (6x + 3y)(14x – 13y)
= 3(2x + y)(14x – 13y)
(iii) 16(5x + 4)2 – 9(3x – 2)2
= (4 (5x + y))2 – (3(3x – 2))2
= [4 (5x + 4) – 3(3x – 2)] [4(5x + 4) + 3(3x – 2)]
[a2 – b2 = (a – b)(a + b)]
= (20x + 16 – 9x + 6][20x + 16 + 9x – 6]
= [11x + 22](29x + 10)
= 11(x + 2)(29x + 10)
(iv) 9x2 – 1/16
= (3x)2 – (1/4)2
= (3x – ¼)(3x + ¼) [(a2 – b2) = (a – b)(a + b)]
(v) 25(x – 2y)2 – 4
= (5 (x – 2y))2 – 22
= [5(x – 2y) – 2][5(x – 2y) + 2]
[a2 – b2 = (a – b)(a + b)]
= (5x – 10y – 2)(5x – 10y + 2)
4. Factorise :
(i) a2 – 23a + 42
(ii) a2 – 23a – 108
(iii) 1 – 18x – 63x2
(iv) 5x2 – 4xy – 12y2
(v) x(3x + 14) + 8
(vi) 5 – 4x(1 + 3x)
(vii) x2y2 – 3xy – 40
(viii) (3x – 2y)2 – 5(3x – 2y) – 24
(ix) 12(a + b)2 – (a + b) – 35
Solution
(i) a2 – 23a + 42
[42 = 21 × 2 and 21 + 2 = 23]
= a2 – 21a – 2a + 42
= a(a – 21) – 2(a – 21)
= (a – 21)(a – 2)
(ii) a2 – 23a – 108
= a2 – 27a + 4a – 108
[27 × 4 = 108 and 27 – 4 = 23]
= 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 – 21x)(1 + 3x)
(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(1 – 2x) + 6x(1 – 2x)
= (1 – 2x)(5 + 6x)
(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
= (3x – 2y)2 – 8(3x – 2y) + 3(3x – 2y) – 24
= (3x – 2y)(3x – 2y – 8) + 3(3x – 2y – 8)
= (3x – 2y – 8)(3x – 2y + 3)
(ix) 12 (a + b)2 – (a + b) – 35
= 12(a + b)2 – 21(a + b) + 20 (a + b) – 35
= 3(a + b) [4(a + b) – 7] + 5[4(a + b) – 7]
= (4a + 4b – 7)(3a + 3b + 5)
5. Factorise :
(i) 15(5x – 4)3 – 10(5x – 4)
(ii) 3a2x – bx + 3a2 – b
(iii) b(c – d)2 + a(d – c) + 3(c – d)
(iv) ax2 + b2y – ab2 - x2y
(v) 1 – 3x – 3y – 4(x + y)2
Solution
(i) 15(5x – 4)3 – 10(5x – 4)
= 5(5x – 4) [3(5x – 4)2 – 2]
= 5(5x – 4)(75x2 – 120x + 46)
(ii) 3a2x – bx + 3a2 – b
= x(3a2 – b) + 1(3a2 – b)
= (x + 1)(3a2 – b)
(iii) b(c – d)2 + a(d – c) + 3(c – d)
= b(c – d)2 – a(c – d) + 3(c – d)
= (c – d)[b(c – d) – a + 3]
= (c – d)(bc – bd – a + 3)
(iv) ax2 + bx2y – ab2 – x2y
= ax2 – ab2 + b2y – x2y
= a(x2 – b2) + y(b2 – x2)
= a(x2 – b2) – y(x2 – b2)
= (x2 – b2)(a – y)
= (x – b)(x + b)(a – y)
(v) 1 – 3x – 3y – 4(x + y)2
= 1 – 3(x + y) – 4(x + y)2
= 1 – 4(x + y) + (x + y) - 4(x + y)2
= 1[1 – 4(x + y)] + (x + y)[1 – 4(x + y)]
= [1 – 4x – 4y](1 + x + y)
6. Factorise:
(i) 2a3 – 50a
(ii) 54a2b2 – 6
(iii) 64a2b – 144b3
(iv) (2x – y)3 – (2x – y)
(v) x2 – 2xy + y2 – z2
(vi) x2 – y2 – 2yz – z2
(vii) 7a5 – 567a
(viii) 5x2 – 20x4/9
Solution
(i) 2a2 – 50a
= 2a(a2 – 25)
= 2a(a2 – 52)
= 2a(a – 5)(a + 5)
(ii) 54a2b2 – 6
= 6(9a2b2 – 1)
= 6[(3ab)2 – (1)2 ]
= 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 – y)[(2x – y)2 – 1]
= (2x – y)(2x – y – 1)(2x – y + 1)
(v) x2 – 2xy + y2 – z2
= (x2 – 2xy + y2) – z2
= (x – y)2 – (z)2
= (x – y – z)(x – y + z)
(vi) x2 – y2 – 2yz – z2
= x2 – (y2 + 2yz + z2)
= x2 – (y + z)2
= (x – y – z)(x + y + z)
(vii) 7a5 – 567a = 7a(a4 – 81)
= 7a (a2)2 – (9)2
= 7a(a2 + 9)(a2 – 9)
= 7a(a2 + 9)((a)2 – (3)2)
= 7a (a2 + 9)(a + 3)(a – 3)
(viii) 5x2 – 20x4/9
= 5x2[1 – 4x2/9]
= 5x2[1 – 2x/3][1 + 2x/3]
7. Factorise xy2 – xz2, Hence, find the value of:
(i) 9 × 82 – 9 × 22
(ii) 40 × 5.52 – 40 × 4.52
Solution
xy2 – xz2
= x(y2 – z2)
= x(y – z)(y + z)
(i) 9 × 82 - 9 × 22
= 9(82 - 22)
= 9(8 - 2) (8 + 2)
= 9 (6)(10)
= 540
(ii) 40 × 5.52 – 40 × 4.52
= 40(5.5)2 – (4.5)2
= 40(5.5 – 4.5)(5.5 + 4.5)
= 40 (1) (10)
= 400
8. Factorise:
(i) (a – 3b)2 – 36b2
(ii) 25(a – 5b)2 – 4(a – 3b)2
(iii) a2 – 0.36b2
(iv) a4 – 625
(v) x4 – 5x2 – 36
(vi) 15(2x – y)2 – 16(2x – y) – 15
Solution
(i) (a – 3b)2 – 36b2
= (a – 3b)2 – (6b)2
= (a – 3b + 6b)(a – 3b – 6b)
= (a + 3b)(a – 9b)
{a2 – b2 = (a + b)(a – b)}
(ii) 25(a – 5b)2 – 4(a – 3b)2
= [5(a – 5b)]2 – [2(a – 3b)]2
= (5a – 25b)2 – (2a – 6b)2
= (5a – 25b + 2a – 6b)(5a – 25b – 2a + 6b)
{a2 – b2 = (a + b)(a – b)}
= (7a – 31b)(3a – 19b)
(iii) a2 – 0.36b2
= (a)2 – (0.6b)2
= (a + 0.6b)(a – 0.6b)
{a2 – b2 = (a + b)(a – b)}
(iv) a4 – 625
= (a2)2 – (25)2
= (a2 + 25)(a2 – 25)
{a2 – b2 = (a + b)(a – b)}
= (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
{∵ - 36 = - 9 × 4, -5 = -9 + 4}
= x2(x2 – 9) + 4(x2 – 9)
= (x2 – 9)(x2 + 4)
= {x2 – (3)2} {x2 + 4}
= (x + 3)(x – 3)(x2 + 4)
= (x2 + 4)(x + 3)(x – 3)
(vi) 15(2x – y)2 – 16(2x – y) – 15
Let 2x – y = a, then
15a2 – 16a – 15
= 15a2 – 25a + 9a – 15
{∵ 15 × (-15) = - 225 ∴ -225 = -25 × 9}
-16 = -25 + 9}
= 5a(3a – 5) + 3(3a – 5)
= (3a – 5)(5a + 3)
= [3(2x – y) – 5][5(2x – y) + 3]
= (6x – 3y – 5) (10x – 5y + 3)
9. Factorise: a2b – b3 Using this result, find the value of 1012 × 100 – 1003.
Solution
a2b – b3
b(a2 – b2)
b(a + b)(a – b)
Now,
1012 × 100 – 1003
= 100(1012 – 1002)
= 100(101 + 100)(101 – 100)
= 100(201)(1)
= 20100
10. Evaluate (using factors) : 3012 x 300 – 3003.
Solution
3012 × 300 – 3003
= 300(3012 – 3002)
= 300(301 + 300)(301 – 300)
= 300(601)(1)
= 180300
(i) 6x3 – 8x2
(ii) 35a3b2c + 42ab2c2
(iii) 36x2y2 – 30x3y3 + 48x3y2
(iv) 8(2a + 3b)3 – 12(2a + 3b)2
(v) 9a(x – 2y)4 – 12a(x – 2y)3
Solution
(i) 6x3 – 8x2 = 2x2(3x – 4)
(ii) 35a3b3c + 42ab2c2
= 7ab2c (5a2 + 6c)
(iii) 36x2y2 – 30x3y3 + 48x3y3
= 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 (3x – 6y – 4)
2. Factorise :
(i) a2 – ab – 3a + 3b
(ii) x2y – xy2 + 5x – 5y
(iii) a2 – ab(1 – b) – b3
(iv) xy2 + (x – 1)y – 1
(v) (ax + by)2 + (bx – ay)2
(vi) ab(x2 + y2) – xy(a2 + b2)
(vii) m – 1 – (m – 1)2 + am – a
Solution
(i) a2 – ab – 3a + 3b
= a(a - b) – 3(a – b)
= (a – b)(a – 3)
(ii) x2y – xy2 + 5x – 5y
= xy(x – y) + 5(x – y)
= (x – y)(xy + 5)
(iii) a2 – ab(1 – ab) – b3
= a2 – ab + ab2 – b3
= a(a – b) + b2(a – b)
= (a – b)(a + b)2
(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)2
= a2x2 + b2y2 + 2abxy + b2x2 + a2y2 – 2abxy
= a2x2 + b2y2 + b2x2 + ay2
= a2x2 + a2y2 + b2x2 + b2y2
= a2(x2 + y2) + b2(x2 + y2)
= (x2 + y2)(a2 + b2)
(vi) ab(x2 + y2) – xy (a2 + b2)
= abx2 + aby2 – a2xy – b2xy
= abx2 – a2xy + aby2 – 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)(1 – m + 1 + a)
= (m – 1)(2 – m + a)
3. Factorise :
(i) a2 – (b – c)2
(ii) 25(2x – y)2 – 16(x – 2y)2
(iii) 16(5x + 4)2 – 9(3x – 2)2
(iv) 9x2 – 1/16
(v) 25(x – 2y)2 – 4
Solution
(i) a2 – (b – c)2
= (a – (b – c))(a + b – c)
[a2 – b2 = (a – b)(a + b)]
= (a – b + c)(a + b – c)
(ii) 25(2x – y)2 – 16(x – 2y)2
= (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]
[a2 – b2 = (a – b)(a + b)]
= (6x + 3y)(14x – 13y)
= 3(2x + y)(14x – 13y)
(iii) 16(5x + 4)2 – 9(3x – 2)2
= (4 (5x + y))2 – (3(3x – 2))2
= [4 (5x + 4) – 3(3x – 2)] [4(5x + 4) + 3(3x – 2)]
[a2 – b2 = (a – b)(a + b)]
= (20x + 16 – 9x + 6][20x + 16 + 9x – 6]
= [11x + 22](29x + 10)
= 11(x + 2)(29x + 10)
(iv) 9x2 – 1/16
= (3x)2 – (1/4)2
= (3x – ¼)(3x + ¼) [(a2 – b2) = (a – b)(a + b)]
(v) 25(x – 2y)2 – 4
= (5 (x – 2y))2 – 22
= [5(x – 2y) – 2][5(x – 2y) + 2]
[a2 – b2 = (a – b)(a + b)]
= (5x – 10y – 2)(5x – 10y + 2)
4. Factorise :
(i) a2 – 23a + 42
(ii) a2 – 23a – 108
(iii) 1 – 18x – 63x2
(iv) 5x2 – 4xy – 12y2
(v) x(3x + 14) + 8
(vi) 5 – 4x(1 + 3x)
(vii) x2y2 – 3xy – 40
(viii) (3x – 2y)2 – 5(3x – 2y) – 24
(ix) 12(a + b)2 – (a + b) – 35
Solution
(i) a2 – 23a + 42
[42 = 21 × 2 and 21 + 2 = 23]
= a2 – 21a – 2a + 42
= a(a – 21) – 2(a – 21)
= (a – 21)(a – 2)
(ii) a2 – 23a – 108
= a2 – 27a + 4a – 108
[27 × 4 = 108 and 27 – 4 = 23]
= 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 – 21x)(1 + 3x)
(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(1 – 2x) + 6x(1 – 2x)
= (1 – 2x)(5 + 6x)
(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
= (3x – 2y)2 – 8(3x – 2y) + 3(3x – 2y) – 24
= (3x – 2y)(3x – 2y – 8) + 3(3x – 2y – 8)
= (3x – 2y – 8)(3x – 2y + 3)
(ix) 12 (a + b)2 – (a + b) – 35
= 12(a + b)2 – 21(a + b) + 20 (a + b) – 35
= 3(a + b) [4(a + b) – 7] + 5[4(a + b) – 7]
= (4a + 4b – 7)(3a + 3b + 5)
5. Factorise :
(i) 15(5x – 4)3 – 10(5x – 4)
(ii) 3a2x – bx + 3a2 – b
(iii) b(c – d)2 + a(d – c) + 3(c – d)
(iv) ax2 + b2y – ab2 - x2y
(v) 1 – 3x – 3y – 4(x + y)2
Solution
(i) 15(5x – 4)3 – 10(5x – 4)
= 5(5x – 4) [3(5x – 4)2 – 2]
= 5(5x – 4)(75x2 – 120x + 46)
(ii) 3a2x – bx + 3a2 – b
= x(3a2 – b) + 1(3a2 – b)
= (x + 1)(3a2 – b)
(iii) b(c – d)2 + a(d – c) + 3(c – d)
= b(c – d)2 – a(c – d) + 3(c – d)
= (c – d)[b(c – d) – a + 3]
= (c – d)(bc – bd – a + 3)
(iv) ax2 + bx2y – ab2 – x2y
= ax2 – ab2 + b2y – x2y
= a(x2 – b2) + y(b2 – x2)
= a(x2 – b2) – y(x2 – b2)
= (x2 – b2)(a – y)
= (x – b)(x + b)(a – y)
(v) 1 – 3x – 3y – 4(x + y)2
= 1 – 3(x + y) – 4(x + y)2
= 1 – 4(x + y) + (x + y) - 4(x + y)2
= 1[1 – 4(x + y)] + (x + y)[1 – 4(x + y)]
= [1 – 4x – 4y](1 + x + y)
6. Factorise:
(i) 2a3 – 50a
(ii) 54a2b2 – 6
(iii) 64a2b – 144b3
(iv) (2x – y)3 – (2x – y)
(v) x2 – 2xy + y2 – z2
(vi) x2 – y2 – 2yz – z2
(vii) 7a5 – 567a
(viii) 5x2 – 20x4/9
Solution
(i) 2a2 – 50a
= 2a(a2 – 25)
= 2a(a2 – 52)
= 2a(a – 5)(a + 5)
(ii) 54a2b2 – 6
= 6(9a2b2 – 1)
= 6[(3ab)2 – (1)2 ]
= 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 – y)[(2x – y)2 – 1]
= (2x – y)(2x – y – 1)(2x – y + 1)
(v) x2 – 2xy + y2 – z2
= (x2 – 2xy + y2) – z2
= (x – y)2 – (z)2
= (x – y – z)(x – y + z)
(vi) x2 – y2 – 2yz – z2
= x2 – (y2 + 2yz + z2)
= x2 – (y + z)2
= (x – y – z)(x + y + z)
(vii) 7a5 – 567a = 7a(a4 – 81)
= 7a (a2)2 – (9)2
= 7a(a2 + 9)(a2 – 9)
= 7a(a2 + 9)((a)2 – (3)2)
= 7a (a2 + 9)(a + 3)(a – 3)
(viii) 5x2 – 20x4/9
= 5x2[1 – 4x2/9]
= 5x2[1 – 2x/3][1 + 2x/3]
7. Factorise xy2 – xz2, Hence, find the value of:
(i) 9 × 82 – 9 × 22
(ii) 40 × 5.52 – 40 × 4.52
Solution
xy2 – xz2
= x(y2 – z2)
= x(y – z)(y + z)
(i) 9 × 82 - 9 × 22
= 9(82 - 22)
= 9(8 - 2) (8 + 2)
= 9 (6)(10)
= 540
(ii) 40 × 5.52 – 40 × 4.52
= 40(5.5)2 – (4.5)2
= 40(5.5 – 4.5)(5.5 + 4.5)
= 40 (1) (10)
= 400
8. Factorise:
(i) (a – 3b)2 – 36b2
(ii) 25(a – 5b)2 – 4(a – 3b)2
(iii) a2 – 0.36b2
(iv) a4 – 625
(v) x4 – 5x2 – 36
(vi) 15(2x – y)2 – 16(2x – y) – 15
Solution
(i) (a – 3b)2 – 36b2
= (a – 3b)2 – (6b)2
= (a – 3b + 6b)(a – 3b – 6b)
= (a + 3b)(a – 9b)
{a2 – b2 = (a + b)(a – b)}
(ii) 25(a – 5b)2 – 4(a – 3b)2
= [5(a – 5b)]2 – [2(a – 3b)]2
= (5a – 25b)2 – (2a – 6b)2
= (5a – 25b + 2a – 6b)(5a – 25b – 2a + 6b)
{a2 – b2 = (a + b)(a – b)}
= (7a – 31b)(3a – 19b)
(iii) a2 – 0.36b2
= (a)2 – (0.6b)2
= (a + 0.6b)(a – 0.6b)
{a2 – b2 = (a + b)(a – b)}
(iv) a4 – 625
= (a2)2 – (25)2
= (a2 + 25)(a2 – 25)
{a2 – b2 = (a + b)(a – b)}
= (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
{∵ - 36 = - 9 × 4, -5 = -9 + 4}
= x2(x2 – 9) + 4(x2 – 9)
= (x2 – 9)(x2 + 4)
= {x2 – (3)2} {x2 + 4}
= (x + 3)(x – 3)(x2 + 4)
= (x2 + 4)(x + 3)(x – 3)
(vi) 15(2x – y)2 – 16(2x – y) – 15
Let 2x – y = a, then
15a2 – 16a – 15
= 15a2 – 25a + 9a – 15
{∵ 15 × (-15) = - 225 ∴ -225 = -25 × 9}
-16 = -25 + 9}
= 5a(3a – 5) + 3(3a – 5)
= (3a – 5)(5a + 3)
= [3(2x – y) – 5][5(2x – y) + 3]
= (6x – 3y – 5) (10x – 5y + 3)
9. Factorise: a2b – b3 Using this result, find the value of 1012 × 100 – 1003.
Solution
a2b – b3
b(a2 – b2)
b(a + b)(a – b)
Now,
1012 × 100 – 1003
= 100(1012 – 1002)
= 100(101 + 100)(101 – 100)
= 100(201)(1)
= 20100
10. Evaluate (using factors) : 3012 x 300 – 3003.
Solution
3012 × 300 – 3003
= 300(3012 – 3002)
= 300(301 + 300)(301 – 300)
= 300(601)(1)
= 180300