FACTORISATION - Questions & Answers

1. Multiple Choice Type :

Choose the correct answer from the options given below.
(a) 4x² + 16 in the form of factors is :
(i) 4(x² + 4)   (ii) 4(x + 2) (x - 2)   (iii) (2x + 4) (2x - 4)   (iv) 16(4x² + 1)
= 4(x²) + 4(4)
= 4(x² + 4)
Answer: (i) 4(x² + 4)

(b) 7(x + 3y)² - 2x - 6y in the form of factors is :
(i) (9x² + 21xy) (x + 6y)   (ii) 9(x + 3y) (x + 3y - 1)   (iii) (x + 3y) (7x + 21y - 2)   (iv) 7(x + 3y) (x + 3y - 3)
= 7(x + 3y)² - 2(x + 3y)
= (x + 3y)[7(x + 3y) - 2]
= (x + 3y)(7x + 21y - 2)
Answer: (iii) (x + 3y) (7x + 21y - 2)

(c) 8(x - y) + 5(y - x) in the form of factors is :
(i) 3(x + y)   (ii) 13(x - y)   (iii) 3(x - y)   (iv) 13(x + y)
= 8(x - y) - 5(x - y)
= (8 - 5)(x - y)
= 3(x - y)
Answer: (iii) 3(x - y)

(d) 4x² + 1/(4x²) - 2 - 6x + 3/(2x) in the form of factors is :
(i) (2x - 1/(2x))(2x - 1/(2x) - 3)   (ii) (2x - 1/(2x))(2x - 1/(2x) + 3)   (iii) 3(2x + 1/(2x))(2x + 1/(2x) - 3)
= (2x)² + (1/(2x))² - 2(2x)(1/(2x)) - 3(2x) + 3(1/(2x))
= (2x - 1/(2x))² - 3(2x - 1/(2x))
= (2x - 1/(2x))(2x - 1/(2x) - 3)
Answer: (i)

(e) 3ab - 6b + 4a² - 8a in the form of factors is :
(i) (a + 2) (4a + 3b)   (ii) (a - 2) (4a + 3b)   (iii) (a - 2) (4a - 3b)   (iv) (a + 2) (4a - 3b)
= 3b(a - 2) + 4a(a - 2)
= (a - 2)(3b + 4a)
= (a - 2)(4a + 3b)
Answer: (ii) (a - 2) (4a + 3b)

Factorise by taking out the common factors :
2. xy(3x² - 2y²) - yz(2y² - 3x²) + zx(15x² - 10y²)
= xy(3x² - 2y²) + yz(3x² - 2y²) + zx[5(3x² - 2y²)]
= xy(3x² - 2y²) + yz(3x² - 2y²) + 5zx(3x² - 2y²)
= (3x² - 2y²)(xy + yz + 5zx)
Answer: (3x² - 2y²)(xy + yz + 5zx)

3. 2x(a - b) + 3y(5a - 5b) + 4z(2b - 2a)
= 2x(a - b) + 15y(a - b) - 8z(a - b)
= (a - b)(2x + 15y - 8z)
Answer: (a - b)(2x + 15y - 8z)

Factorise by grouping method :
4. 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)
Answer: 4(a + b)(4a + 4b - 1)

5. a⁴ - 2a³ - 4a + 8
= a³(a - 2) - 4(a - 2)
= (a - 2)(a³ - 4)
Answer: (a - 2)(a³ - 4)

6. ab(x² + 1) + x(a² + b²)
= abx² + ab + a²x + b²x
= ax(bx + a) + b(a + bx)
= (bx + a)(ax + b)
Answer: (bx + a)(ax + b)

7. (ax + by)² + (bx - ay)²
= a²x² + 2abxy + b²y² + b²x² - 2abxy + a²y²
= a²x² + b²y² + b²x² + a²y²
= x²(a² + b²) + y²(a² + b²)
= (a² + b²)(x² + y²)
Answer: (a² + b²)(x² + y²)

8. a²x² + (ax² + 1)x + a
= a²x² + ax³ + x + a
= ax²(a + x) + 1(x + a)
= (a + x)(ax² + 1)
Answer: (a + x)(ax² + 1)

9. y² - (a + b)y + ab
= y² - ay - by + ab
= y(y - a) - b(y - a)
= (y - a)(y - b)
Answer: (y - a)(y - b)

EXERCISE 5 (B)
1. Multiple Choice Type :
Choose the correct answer from the options given below.
(a) x² + 2(5x + 12) in the form of factors is :
(i) (x - 6)(x + 4)   (ii) (x - 6)(x - 4)   (iii) (x + 6)(x + 4)   (iv) (x + 6)(x - 4)
= x² + 10x + 24
= x² + 6x + 4x + 24
= x(x + 6) + 4(x + 6)
= (x + 6)(x + 4)
Answer: (iii) (x + 6)(x + 4)

(b) x(2x - 5) + 3 in the form of factors is :
(i) (x + 1)(2x - 3)   (ii) (x + 1)(2x + 3)   (iii) (x - 1)(2x + 3)   (iv) (x - 1)(2x - 3)
= 2x² - 5x + 3
= 2x² - 2x - 3x + 3
= 2x(x - 1) - 3(x - 1)
= (x - 1)(2x - 3)
Answer: (iv) (x - 1)(2x - 3)

(c) 6x² - 2 - 4x in the form of factors is :
(i) 2(x - 1)(3x - 1)   (ii) 2(x - 1)(3x + 1)   (iii) 2(x - 1)(3x - 1)   (iv) (6x - 2)(x + 1)
= 2(3x² - 2x - 1)
= 2(3x² - 3x + x - 1)
= 2[3x(x - 1) + 1(x - 1)]
= 2(x - 1)(3x + 1)
Answer: (ii) 2(x - 1)(3x + 1)

(d) x(x - 1) - 20 in the form of factors is :
(i) (x - 5)(x + 4)   (ii) (x - 5)(x - 4)   (iii) (x + 5)(x - 4)   (iv) (x + 5)(x + 4)
= x² - x - 20
= x² - 5x + 4x - 20
= x(x - 5) + 4(x - 5)
= (x - 5)(x + 4)
Answer: (i) (x - 5)(x + 4)

(e) 5 - x(6 - x) in the form of factors is :
(i) (1 - x)(5 + x)   (ii) (1 + x)(5 + x)   (iii) (1 - x)(5 - x)   (iv) (1 + x)(5 - x)
= 5 - 6x + x²
= x² - 6x + 5
= x² - 5x - x + 5
= x(x - 5) - 1(x - 5)
= (x - 1)(x - 5)
= -(1 - x) * -(5 - x)
= (1 - x)(5 - x)
Answer: (iii) (1 - x)(5 - x)

Factorise :
2. 1 - 2a - 3a²
= 1 - 3a + a - 3a²
= 1(1 - 3a) + a(1 - 3a)
= (1 - 3a)(1 + a)
Answer: (1 - 3a)(1 + a)

3. 24a³ + 37a² - 5a
= a(24a² + 37a - 5)
= a(24a² + 40a - 3a - 5)
= a[8a(3a + 5) - 1(3a + 5)]
= a(3a + 5)(8a - 1)
Answer: a(3a + 5)(8a - 1)

4. 3 - a(4 + 7a)
= 3 - 4a - 7a²
= 3 + 3a - 7a - 7a²
= 3(1 + a) - 7a(1 + a)
= (1 + a)(3 - 7a)
Answer: (1 + a)(3 - 7a)

5. (2a + b)² - 6a - 3b - 4
= (2a + b)² - 3(2a + b) - 4
Let x = (2a + b)
= x² - 3x - 4
= x² - 4x + x - 4
= (x - 4)(x + 1)
= (2a + b - 4)(2a + b + 1)
Answer: (2a + b - 4)(2a + b + 1)

6. 1 - 2a - 2b - 3(a + b)²
= 1 - 2(a + b) - 3(a + b)²
Let x = (a + b)
= 1 - 2x - 3x²
= 1 - 3x + x - 3x²
= 1(1 - 3x) + x(1 - 3x)
= (1 - 3x)(1 + x)
= (1 - 3(a + b))(1 + a + b)
= (1 - 3a - 3b)(1 + a + b)
Answer: (1 - 3a - 3b)(1 + a + b)

7. 3a² - 1 - 2a
= 3a² - 2a - 1
= 3a² - 3a + a - 1
= 3a(a - 1) + 1(a - 1)
= (a - 1)(3a + 1)
Answer: (a - 1)(3a + 1)

8. x² + 3x + 2 + ax + 2a
= (x² + 2x + x + 2) + a(x + 2)
= [x(x + 2) + 1(x + 2)] + a(x + 2)
= (x + 2)(x + 1) + a(x + 2)
= (x + 2)(x + 1 + a)
Answer: (x + 2)(x + 1 + a)

9. (3x - 2y)² + 3(3x - 2y) - 10
Let a = (3x - 2y)
= a² + 3a - 10
= a² + 5a - 2a - 10
= a(a + 5) - 2(a + 5)
= (a + 5)(a - 2)
= (3x - 2y + 5)(3x - 2y - 2)
Answer: (3x - 2y + 5)(3x - 2y - 2)

10. 5 - (3a² - 2a)(6 - 3a² + 2a)
= 5 - (3a² - 2a)[6 - (3a² - 2a)]
Let x = (3a² - 2a)
= 5 - x(6 - x)
= 5 - 6x + x²
= x² - 6x + 5
= (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)(a - 1)(3a + 1)
Answer: (3a - 5)(a + 1)(a - 1)(3a + 1)

EXERCISE 5 (C)
1. Multiple Choice Type :
Choose the correct answer from the options given below.
(a) x⁴ - 1 in the form of factors is :
(i) (x + 1)(x - 1)(x² - 1)   (ii) (x - 1)(x + 1)(x² + 1)   (iii) (x - 1)(x - 2)(x² + 2)   (iv) (x + 1)(x + 2)(x² - 1)
= (x²)² - 1²
= (x² - 1)(x² + 1)
= (x - 1)(x + 1)(x² + 1)
Answer: (ii) (x - 1)(x + 1)(x² + 1)

(b) 2a² - 18 in the form of factors is :
(i) 2(a + 3)(a - 3)   (ii) 2(a + 1)(a - 9)   (iii) 2(a - 1)(a - 9)   (iv) (2a - 1)(a - 9)
= 2(a² - 9)
= 2(a² - 3²)
= 2(a - 3)(a + 3)
Answer: (i) 2(a + 3)(a - 3)

(c) 27x³ - 48x in the form of factors is :
(i) 3(3x - 4)(3x - 4)   (ii) 3x(3x + 4)(3x + 4)   (iii) 3(3x² - 4x)(3x + 4)   (iv) 3x(3x - 4)(3x + 4)
= 3x(9x² - 16)
= 3x((3x)² - 4²)
= 3x(3x - 4)(3x + 4)
Answer: (iv) 3x(3x - 4)(3x + 4)

(d) x² - (x - 4y)² in the form of factors is :
(i) 8y(x - 2y)   (ii) 8y(x + 2y)   (iii) 4y(x + 2y)   (iv) 4y(x - 2y)
= [x - (x - 4y)][x + (x - 4y)]
= [x - x + 4y][2x - 4y]
= 4y(2x - 4y)
= 4y * 2(x - 2y)
= 8y(x - 2y)
Answer: (i) 8y(x - 2y)

(e) 9x² + 1/(9x²) + 1 in the form of factors is :
(i) (3x + 1/(3x) + 1)(3x + 1/(3x) - 1)
= 9x² + 1/(9x²) + 2 - 1
= (3x)² + (1/(3x))² + 2(3x)(1/(3x)) - 1²
= (3x + 1/(3x))² - 1²
= (3x + 1/(3x) + 1)(3x + 1/(3x) - 1)
Answer: (i)

Factorise :
2. a² - (2a + 3b)²
= [a - (2a + 3b)][a + (2a + 3b)]
= (a - 2a - 3b)(a + 2a + 3b)
= (-a - 3b)(3a + 3b)
= -3(a + b)(a + 3b)
Answer: -3(a + b)(a + 3b)

3. 25(2a - b)² - 81b²
= [5(2a - b)]² - (9b)²
= [5(2a - b) - 9b][5(2a - b) + 9b]
= (10a - 5b - 9b)(10a - 5b + 9b)
= (10a - 14b)(10a + 4b)
= 2(5a - 7b) * 2(5a + 2b)
= 4(5a - 7b)(5a + 2b)
Answer: 4(5a - 7b)(5a + 2b)

4. 50a³ - 2a
= 2a(25a² - 1)
= 2a((5a)² - 1²)
= 2a(5a - 1)(5a + 1)
Answer: 2a(5a - 1)(5a + 1)

5. 4a²b - 9b³
= b(4a² - 9b²)
= b((2a)² - (3b)²)
= b(2a - 3b)(2a + 3b)
Answer: b(2a - 3b)(2a + 3b)

6. 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)]
= (3a - 6 - 4a - 8)(3a - 6 + 4a + 8)
= (-a - 14)(7a + 2)
= -(a + 14)(7a + 2)
Answer: -(a + 14)(7a + 2)

7. (a + b)³ - a - b
= (a + b)³ - (a + b)
= (a + b)[(a + b)² - 1]
= (a + b)(a + b - 1)(a + b + 1)
Answer: (a + b)(a + b - 1)(a + b + 1)

8. a(a - 1) - b(b - 1)
= a² - a - b² + b
= (a² - b²) - (a - b)
= (a - b)(a + b) - 1(a - b)
= (a - b)(a + b - 1)
Answer: (a - b)(a + b - 1)

9. 4a² - (4b² + 4bc + c²)
= 4a² - (2b + c)²
= (2a)² - (2b + c)²
= (2a - (2b + c))(2a + (2b + c))
= (2a - 2b - c)(2a + 2b + c)
Answer: (2a - 2b - c)(2a + 2b + c)

10. 4a² - 49b² + 2a - 7b
= (2a)² - (7b)² + 1(2a - 7b)
= (2a - 7b)(2a + 7b) + 1(2a - 7b)
= (2a - 7b)(2a + 7b + 1)
Answer: (2a - 7b)(2a + 7b + 1)

11. 4a² - 12a + 9 - 49b²
= (2a - 3)² - (7b)²
= (2a - 3 - 7b)(2a - 3 + 7b)
Answer: (2a - 3 - 7b)(2a - 3 + 7b)

12. 4xy - x² - 4y² + z²
= z² - (x² - 4xy + 4y²)
= z² - (x - 2y)²
= (z - (x - 2y))(z + (x - 2y))
= (z - x + 2y)(z + x - 2y)
Answer: (z - x + 2y)(z + x - 2y)

13. a² + b² - c² - d² + 2ab - 2cd
= (a² + 2ab + b²) - (c² + 2cd + d²)
= (a + b)² - (c + d)²
= (a + b - (c + d))(a + b + (c + d))
= (a + b - c - d)(a + b + c + d)
Answer: (a + b - c - d)(a + b + c + d)

14. 4x² - 12ax - y² - z² - 2yz + 9a²
= (4x² - 12ax + 9a²) - (y² + 2yz + z²)
= (2x - 3a)² - (y + z)²
= (2x - 3a - (y + z))(2x - 3a + (y + z))
= (2x - 3a - y - z)(2x - 3a + y + z)
Answer: (2x - 3a - y - z)(2x - 3a + y + z)

15. (a² - 1)(b² - 1) + 4ab
= a²b² - a² - b² + 1 + 4ab
= a²b² + 2ab + 1 - a² + 2ab - b²
= (a²b² + 2ab + 1) - (a² - 2ab + b²)
= (ab + 1)² - (a - b)²
= (ab + 1 - (a - b))(ab + 1 + (a - b))
= (ab - a + b + 1)(ab + a - b + 1)
Answer: (ab - a + b + 1)(ab + a - b + 1)

16. x⁴ + x² + 1
= x⁴ + 2x² + 1 - x²
= (x² + 1)² - x²
= (x² + 1 - x)(x² + 1 + x)
= (x² - x + 1)(x² + x + 1)
Answer: (x² - x + 1)(x² + x + 1)

17. (a² + b² - 4c²)² - 4a²b²
= (a² + b² - 4c²)² - (2ab)²
= (a² + b² - 4c² - 2ab)(a² + b² - 4c² + 2ab)
= ((a - b)² - 4c²)((a + b)² - 4c²)
= (a - b - 2c)(a - b + 2c)(a + b - 2c)(a + b + 2c)
Answer: (a - b - 2c)(a - b + 2c)(a + b - 2c)(a + b + 2c)

18. (x² + 4y² - 9z²)² - 16x²y²
= (x² + 4y² - 9z²)² - (4xy)²
= (x² + 4y² - 9z² - 4xy)(x² + 4y² - 9z² + 4xy)
= ((x - 2y)² - 9z²)((x + 2y)² - 9z²)
= (x - 2y - 3z)(x - 2y + 3z)(x + 2y - 3z)(x + 2y + 3z)
Answer: (x - 2y - 3z)(x - 2y + 3z)(x + 2y - 3z)(x + 2y + 3z)

19. (a + b)² - a² + b²
= (a + b)² - (a² - b²)
= (a + b)² - (a - b)(a + b)
= (a + b)[(a + b) - (a - b)]
= (a + b)[a + b - a + b]
= (a + b)(2b)
= 2b(a + b)
Answer: 2b(a + b)

20. a² - b² - (a + b)²
= (a - b)(a + b) - (a + b)²
= (a + b)[(a - b) - (a + b)]
= (a + b)[a - b - a - b]
= (a + b)(-2b)
= -2b(a + b)
Answer: -2b(a + b)

EXERCISE 5 (D)
1. Multiple Choice Type :
Choose the correct answer from the options given below.
(a) a³ - 8 in the form of factors is :
(i) (a - 2) (a² + 2a + 4)
= a³ - 2³
= (a - 2)(a² + 2a + 2²)
= (a - 2)(a² + 2a + 4)
Answer: (i)

(b) 27 + 8x³ in the form of factors is :
(ii) (3 + 2x) (9 - 6x + 4x²)
= 3³ + (2x)³
= (3 + 2x)(3² - 3(2x) + (2x)²)
= (3 + 2x)(9 - 6x + 4x²)
Answer: (ii)

(c) 8a³ + 1 in the form of factors is :
(i) (2a + 1) (4a² - 2a + 1)
= (2a)³ + 1³
= (2a + 1)((2a)² - 2a(1) + 1²)
= (2a + 1)(4a² - 2a + 1)
Answer: (i)

(d) x³/8 - y³/27 in the form of factors is :
(i) (x/2 - y/3)(x²/4 + xy/6 + y²/9)
= (x/2)³ - (y/3)³
= (x/2 - y/3)((x/2)² + (x/2)(y/3) + (y/3)²)
= (x/2 - y/3)(x²/4 + xy/6 + y²/9)
Answer: (i)

Factorise :
2. 64 - a³b³
= 4³ - (ab)³
= (4 - ab)(4² + 4(ab) + (ab)²)
= (4 - ab)(16 + 4ab + a²b²)
Answer: (4 - ab)(16 + 4ab + a²b²)

3. a⁶ + 27b³
= (a²)³ + (3b)³
= (a² + 3b)((a²)² - a²(3b) + (3b)²)
= (a² + 3b)(a⁴ - 3a²b + 9b²)
Answer: (a² + 3b)(a⁴ - 3a²b + 9b²)

4. 3x⁷y - 81x⁴y⁴
= 3x⁴y(x³ - 27y³)
= 3x⁴y(x³ - (3y)³)
= 3x⁴y(x - 3y)(x² + x(3y) + (3y)²)
= 3x⁴y(x - 3y)(x² + 3xy + 9y²)
Answer: 3x⁴y(x - 3y)(x² + 3xy + 9y²)

5. a³ - 27/a³
= a³ - (3/a)³
= (a - 3/a)(a² + a(3/a) + (3/a)²)
= (a - 3/a)(a² + 3 + 9/a²)
Answer: (a - 3/a)(a² + 3 + 9/a²)

6. a³ + 0.064
= a³ + (0.4)³
= (a + 0.4)(a² - 0.4a + (0.4)²)
= (a + 0.4)(a² - 0.4a + 0.16)
Answer: (a + 0.4)(a² - 0.4a + 0.16)

7. (x - y)³ - 8x³
= (x - y)³ - (2x)³
= ((x - y) - 2x)((x - y)² + (x - y)(2x) + (2x)²)
= (-x - y)(x² - 2xy + y² + 2x² - 2xy + 4x²)
= -(x + y)(7x² - 4xy + y²)
Answer: -(x + y)(7x² - 4xy + y²)

8. 8a³/27 - b³/8
= (2a/3)³ - (b/2)³
= (2a/3 - b/2)((2a/3)² + (2a/3)(b/2) + (b/2)²)
= (2a/3 - b/2)(4a²/9 + ab/3 + b²/4)
Answer: (2a/3 - b/2)(4a²/9 + ab/3 + b²/4)

9. a⁶ - b⁶
= (a³)² - (b³)²
= (a³ - b³)(a³ + b³)
= (a - b)(a² + ab + b²)(a + b)(a² - ab + b²)
Answer: (a - b)(a + b)(a² + ab + b²)(a² - ab + b²)

10. a⁶ - 7a³ - 8
Let x = a³
= x² - 7x - 8
= (x - 8)(x + 1)
= (a³ - 8)(a³ + 1)
= (a - 2)(a² + 2a + 4)(a + 1)(a² - a + 1)
Answer: (a - 2)(a + 1)(a² + 2a + 4)(a² - a + 1)

11. a³ - 27b³ + 2a²b - 6ab²
= (a³ - (3b)³) + 2ab(a - 3b)
= (a - 3b)(a² + 3ab + 9b²) + 2ab(a - 3b)
= (a - 3b)(a² + 3ab + 9b² + 2ab)
= (a - 3b)(a² + 5ab + 9b²)
Answer: (a - 3b)(a² + 5ab + 9b²)

12. 8a³ - b³ - 4ax + 2bx
= (2a)³ - b³ - 2x(2a - b)
= (2a - b)(4a² + 2ab + b²) - 2x(2a - b)
= (2a - b)(4a² + 2ab + b² - 2x)
Answer: (2a - b)(4a² + 2ab + b² - 2x)

13. a - b - a³ + b³
= (a - b) - (a³ - b³)
= (a - b) - (a - b)(a² + ab + b²)
= (a - b)(1 - (a² + ab + b²))
= (a - b)(1 - a² - ab - b²)
Answer: (a - b)(1 - a² - ab - b²)

EXERCISE 5 (E)
1. Multiple Choice Type :
Choose the correct answer from the options given below.
(a) x⁴ + 1/x⁴ - 2 in the form of factors is :
(i) (x - 1/x)² (x + 1/x)²
= (x²)² + (1/x²)² - 2(x²)(1/x²)
= (x² - 1/x²)²
= [(x - 1/x)(x + 1/x)]²
= (x - 1/x)²(x + 1/x)²
Answer: (i)

(b) x² + 1/x² - 3 in the form of factors is :
(iv) (x - 1/x + 1)(x - 1/x - 1)
= x² + 1/x² - 2 - 1
= (x - 1/x)² - 1²
= (x - 1/x + 1)(x - 1/x - 1)
Answer: (iv)

Factorise :
2. x² + 1/(4x²) + 1 - 7x - 7/(2x)
= (x² + 2(x)(1/(2x)) + 1/(4x²)) - 7(x + 1/(2x))
= (x + 1/(2x))² - 7(x + 1/(2x))
= (x + 1/(2x))(x + 1/(2x) - 7)
Answer: (x + 1/(2x))(x + 1/(2x) - 7)

3. 9a² + 1/(9a²) - 2 - 12a + 4/(3a)
= (3a)² + 1/(3a)² - 2(3a)(1/(3a)) - 4(3a - 1/(3a))
= (3a - 1/(3a))² - 4(3a - 1/(3a))
= (3a - 1/(3a))(3a - 1/(3a) - 4)
Answer: (3a - 1/(3a))(3a - 1/(3a) - 4)

4. x² + ((a² + 1)/a)x + 1
= x² + ax + (1/a)x + 1
= x(x + a) + (1/a)(x + a)
= (x + a)(x + 1/a)
Answer: (x + a)(x + 1/a)

5. x⁴ + y⁴ - 27x²y²
= x⁴ + y⁴ - 2x²y² - 25x²y²
= (x² - y²)² - (5xy)²
= (x² - y² - 5xy)(x² - y² + 5xy)
Answer: (x² - y² - 5xy)(x² - y² + 5xy)

6. 4x⁴ + 9y⁴ + 11x²y²
= (2x²)² + (3y²)² + 12x²y² - x²y²
= (2x² + 3y²)² - (xy)²
= (2x² + 3y² - xy)(2x² + 3y² + xy)
Answer: (2x² + 3y² - xy)(2x² + 3y² + xy)

7. x² + 1/x² - 3
= x² + 1/x² - 2 - 1
= (x - 1/x)² - 1²
= (x - 1/x - 1)(x - 1/x + 1)
Answer: (x - 1/x - 1)(x - 1/x + 1)

8. a - b - 4a² + 4b²
= (a - b) - 4(a² - b²)
= (a - b) - 4(a - b)(a + b)
= (a - b)[1 - 4(a + b)]
= (a - b)(1 - 4a - 4b)
Answer: (a - b)(1 - 4a - 4b)

9. (2a - 3)² - 2(2a - 3)(a - 1) + (a - 1)²
= [(2a - 3) - (a - 1)]²
= (2a - 3 - a + 1)²
= (a - 2)²
Answer: (a - 2)²

10. (a² - 3a)(a² - 3a + 7) + 10
Let x = a² - 3a
= x(x + 7) + 10
= x² + 7x + 10
= (x + 5)(x + 2)
= (a² - 3a + 5)(a² - 3a + 2)
= (a² - 3a + 5)(a - 2)(a - 1)
Answer: (a² - 3a + 5)(a - 2)(a - 1)

11. (a² - a)(4a² - 4a - 5) - 6
Let x = a² - a
= x(4x - 5) - 6
= 4x² - 5x - 6
= 4x² - 8x + 3x - 6
= 4x(x - 2) + 3(x - 2)
= (x - 2)(4x + 3)
= (a² - a - 2)(4(a² - a) + 3)
= (a - 2)(a + 1)(4a² - 4a + 3)
Answer: (a - 2)(a + 1)(4a² - 4a + 3)

12. x⁴ + y⁴ - 3x²y²
= x⁴ + y⁴ - 2x²y² - x²y²
= (x² - y²)² - (xy)²
= (x² - y² - xy)(x² - y² + xy)
Answer: (x² - y² - xy)(x² - y² + xy)

13. 5a² - b² - 4ab + 7a - 7b
= 4a² - 4ab + a² - b² + 7(a - b)
= 4a(a - b) + (a - b)(a + b) + 7(a - b)
= (a - b)(4a + a + b + 7)
= (a - b)(5a + b + 7)
Answer: (a - b)(5a + b + 7)

14. 12(3x - 2y)² - 3x + 2y - 1
= 12(3x - 2y)² - 1(3x - 2y) - 1
Let a = 3x - 2y
= 12a² - a - 1
= 12a² - 4a + 3a - 1
= 4a(3a - 1) + 1(3a - 1)
= (3a - 1)(4a + 1)
= (3(3x - 2y) - 1)(4(3x - 2y) + 1)
= (9x - 6y - 1)(12x - 8y + 1)
Answer: (9x - 6y - 1)(12x - 8y + 1)

15. 4(2x - 3y)² - 8x + 12y - 3
= 4(2x - 3y)² - 4(2x - 3y) - 3
Let a = 2x - 3y
= 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: (4x - 6y - 3)(4x - 6y + 1)

16. 3 - 5x + 5y - 12(x - y)²
= 3 - 5(x - y) - 12(x - y)²
Let a = x - y
= 3 - 5a - 12a²
= 3 - 9a + 4a - 12a²
= 3(1 - 3a) + 4a(1 - 3a)
= (1 - 3a)(3 + 4a)
= (1 - 3(x - y))(3 + 4(x - y))
= (1 - 3x + 3y)(3 + 4x - 4y)
Answer: (1 - 3x + 3y)(3 + 4x - 4y)

TEST YOURSELF
1. Multiple Choice Type :
Choose the correct answer from the options given below.
(a) ((a + b)(a² - ab + b²))/(a²b + ab²) in simplest form is equal to :
= (a³ + b³) / (ab(a + b))
= ((a + b)(a² - ab + b²)) / (ab(a + b))
= (a² - ab + b²) / ab
= a/b - 1 + b/a
Answer: (i) a/b - 1 + b/a

(b) L.C.M. of x² + 3x + 2 and x² - 2x - 3 in simplest form is :
x² + 3x + 2 = (x + 1)(x + 2)
x² - 2x - 3 = (x - 3)(x + 1)
L.C.M. = (x + 1)(x + 2)(x - 3)
Answer: (i) (x + 1)(x + 2)(x - 3)

(c) H.C.F. of x² + 3x + 2 and x² - 2x - 3 is :
x² + 3x + 2 = (x + 1)(x + 2)
x² - 2x - 3 = (x - 3)(x + 1)
H.C.F. = (x + 1)
Answer: (i) (x + 1)

(d) (3a - 1)² - 6a + 2 is equal to :
= (3a - 1)² - 2(3a - 1)
= (3a - 1)(3a - 1 - 2)
= (3a - 1)(3a - 3)
= 3(3a - 1)(a - 1)
Answer: (ii) 3(3a - 1)(a - 1)

(e) (1/3)x² - 2x - 9 is equal to :
= (1/3)(x² - 6x - 27)
= (1/3)(x - 9)(x + 3)
Answer: (i) (1/3)(x - 9)(x + 3)

(f) (x² + 3x) men can do a piece of work in (x² - 2x) days, then one day work of 1 man is :
Total man-days = (x² + 3x)(x² - 2x)
One day work of 1 man = 1 / ((x² + 3x)(x² - 2x))
This does not match options (i), (ii), (iii).
Answer: (iv) none of these

(g) Statement (1) : (x³ - x) is spent in buying some identical articles at (x - 1) each. Number of articles bought = (x³ - x) / (x - 1)
Statement (2) : The number of articles bought = (x³ - x) / (x - 1) = (x(x + 1)(x - 1)) / (x - 1) = x² + x
Both statements are mathematically sound and correct.
Answer: (i) Both the statements are true.

(h) Statement (1) : The area of a rectangle is x² - 5x + 6 and the longer side of the rectangle is (x - 2).
Statement (2) : x² - 5x + 6 = (x - 2)(x - 3) => for every positive value of x (x > 3), (x - 2) is greater.
Statement 2 correctly justifies Statement 1. Since (x - 2) > (x - 3) for x > 3.
Answer: (i) Both the statements are true.

(i) Assertion (A) : Distance of (x² - 7x + 12) km is covered in (x² - 16) hrs. Speed = (x² - 7x + 12)/(x² - 16) km/h
Reason (R) : Speed = Distance * Time = (x² - 7x + 12)(x² - 16) km/h
Speed is Distance / Time. Reason (R) states Speed = Distance * Time, which is false.
Answer: (i) A is true, R is false.

Factorise :
2. a² + 1/a² - 2 - 3a + 3/a
= (a - 1/a)² - 3(a - 1/a)
= (a - 1/a)(a - 1/a - 3)
Answer: (a - 1/a)(a - 1/a - 3)

3. x² + y² + x + y + 2xy
= (x² + 2xy + y²) + (x + y)
= (x + y)² + 1(x + y)
= (x + y)(x + y + 1)
Answer: (x + y)(x + y + 1)

4. a² + 4b² - 3a + 6b - 4ab
= (a² - 4ab + 4b²) - 3(a - 2b)
= (a - 2b)² - 3(a - 2b)
= (a - 2b)(a - 2b - 3)
Answer: (a - 2b)(a - 2b - 3)

5. m(x - 3y)² + n(3y - x) + 5x - 15y
= m(x - 3y)² - n(x - 3y) + 5(x - 3y)
= (x - 3y)(m(x - 3y) - n + 5)
= (x - 3y)(mx - 3my - n + 5)
Answer: (x - 3y)(mx - 3my - n + 5)

6. x(6x - 5y) - 4(6x - 5y)²
= (6x - 5y)[x - 4(6x - 5y)]
= (6x - 5y)(x - 24x + 20y)
= (6x - 5y)(20y - 23x)
Answer: (6x - 5y)(20y - 23x)

7. 1/35 + 12/35 a + a²
= (1/35)(1 + 12a + 35a²)
= (1/35)(35a² + 5a + 7a + 1)
= (1/35)[5a(7a + 1) + 1(7a + 1)]
= (1/35)(7a + 1)(5a + 1)
Answer: (1/35)(7a + 1)(5a + 1)

8. (x² - 3x)(x² - 3x - 1) - 20
Let a = x² - 3x
= a(a - 1) - 20
= a² - a - 20
= (a - 5)(a + 4)
= (x² - 3x - 5)(x² - 3x + 4)
Answer: (x² - 3x - 5)(x² - 3x + 4)

9. For each trinomial (quadratic expression), given below, find whether it is factorisable or not. Factorise, if possible.
(i) x² - 3x - 54
b² - 4ac = (-3)² - 4(1)(-54) = 9 + 216 = 225 = 15² (Perfect square, so factorisable)
= x² - 9x + 6x - 54
= x(x - 9) + 6(x - 9)
= (x - 9)(x + 6)
Answer: Factorisable, (x - 9)(x + 6)

(ii) 2x² - 7x - 15
b² - 4ac = (-7)² - 4(2)(-15) = 49 + 120 = 169 = 13² (Factorisable)
= 2x² - 10x + 3x - 15
= 2x(x - 5) + 3(x - 5)
= (x - 5)(2x + 3)
Answer: Factorisable, (x - 5)(2x + 3)

(iii) 2x² + 2x - 75
b² - 4ac = (2)² - 4(2)(-75) = 4 + 600 = 604 (Not a perfect square)
Answer: Not factorisable.

(iv) 3x² + 4x - 10
b² - 4ac = (4)² - 4(3)(-10) = 16 + 120 = 136 (Not a perfect square)
Answer: Not factorisable.

(v) x(2x - 1) - 1
= 2x² - x - 1
b² - 4ac = (-1)² - 4(2)(-1) = 1 + 8 = 9 = 3² (Factorisable)
= 2x² - 2x + x - 1
= 2x(x - 1) + 1(x - 1)
= (x - 1)(2x + 1)
Answer: Factorisable, (x - 1)(2x + 1)

10. Factorise :
(i) 4√3x² + 5x - 2√3
= 4√3x² + 8x - 3x - 2√3
= 4x(√3x + 2) - √3(√3x + 2)
= (√3x + 2)(4x - √3)
Answer: (√3x + 2)(4x - √3)

(ii) 7√2x² - 10x - 4√2
= 7√2x² - 14x + 4x - 4√2
= 7√2x(x - √2) + 4(x - √2)
= (x - √2)(7√2x + 4)
Answer: (x - √2)(7√2x + 4)

11. Give possible expressions for the length and the breadth of the rectangle whose area is 12x² - 35x + 25
= 12x² - 15x - 20x + 25
= 3x(4x - 5) - 5(4x - 5)
= (4x - 5)(3x - 5)
Answer: Length and Breadth can be (4x - 5) and (3x - 5)

12. 9a² - (a² - 4)²
= (3a)² - (a² - 4)²
= (3a - (a² - 4))(3a + (a² - 4))
= (4 + 3a - a²)(a² + 3a - 4)
= -(a² - 3a - 4)(a² + 3a - 4)
= -(a - 4)(a + 1)(a + 4)(a - 1)
= (4 - a)(a + 1)(a + 4)(a - 1)
Answer: (4 - a)(a + 1)(a + 4)(a - 1)

13. x² + 1/x² - 11
= x² + 1/x² - 2 - 9
= (x - 1/x)² - 3²
= (x - 1/x - 3)(x - 1/x + 3)
Answer: (x - 1/x - 3)(x - 1/x + 3)

14. 4x² + 1/(4x²) + 1
= 4x² + 1/(4x²) + 2 - 1
= (2x + 1/(2x))² - 1²
= (2x + 1/(2x) - 1)(2x + 1/(2x) + 1)
Answer: (2x + 1/(2x) - 1)(2x + 1/(2x) + 1)

15. 4x⁴ - x² - 12x - 36
= 4x⁴ - (x² + 12x + 36)
= (2x²)² - (x + 6)²
= (2x² - (x + 6))(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)
Answer: (x - 2)(2x + 3)(2x² + x + 6)

16. 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)
Answer: (b + c)(a - b - c)(a + b + c)

17. 2x³ + 54y³ - 4x - 12y
= 2(x³ + 27y³) - 4(x + 3y)
= 2((x)³ + (3y)³) - 4(x + 3y)
= 2(x + 3y)(x² - 3xy + 9y²) - 4(x + 3y)
= 2(x + 3y)(x² - 3xy + 9y² - 2)
Answer: 2(x + 3y)(x² - 3xy + 9y² - 2)

18. 1029 - 3x³
= 3(343 - x³)
= 3(7³ - x³)
= 3(7 - x)(49 + 7x + x²)
Answer: 3(7 - x)(49 + 7x + x²)

19. Show that :
(i) 13³ - 5³ is divisible by 8.
= (13 - 5)(13² + 13 * 5 + 5²)
= 8 * (169 + 65 + 25)
Since 8 is a factor, the expression is entirely divisible by 8.
Answer: Shown.

(ii) 35³ + 27³ is divisible by 62.
= (35 + 27)(35² - 35 * 27 + 27²)
= 62 * (1225 - 945 + 729)
Since 62 is a factor, the expression is entirely divisible by 62.
Answer: Shown.

20. 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)
Let a = 5.67 and b = 4.33
= (a³ + b³) / (a² - ab + b²)
= ((a + b)(a² - ab + b²)) / (a² - ab + b²)
= a + b
= 5.67 + 4.33
= 10.00
Answer: 10

Factorise :
21. 9x² + 3x - 8y - 64y²
= (9x² - 64y²) + (3x - 8y)
= (3x - 8y)(3x + 8y) + 1(3x - 8y)
= (3x - 8y)(3x + 8y + 1)
Answer: (3x - 8y)(3x + 8y + 1)

22. 2√3x² + x - 5√3
= 2√3x² + 6x - 5x - 5√3
= 2√3x(x + √3) - 5(x + √3)
= (x + √3)(2√3x - 5)
Answer: (x + √3)(2√3x - 5)

23. (1/4)(a + b)² - (9/16)(2a - b)²
= [1/2 (a + b)]² - [3/4 (2a - b)]²
= [1/2 (a + b) - 3/4 (2a - b)][1/2 (a + b) + 3/4 (2a - b)]
= [(2a + 2b - 6a + 3b) / 4][(2a + 2b + 6a - 3b) / 4]
= ((5b - 4a)/4) * ((8a - b)/4)
= (5b - 4a)(8a - b) / 16
Answer: (5b - 4a)(8a - b) / 16

24. 2(ab + cd) - a² - b² + c² + d²
= (c² + 2cd + d²) - (a² - 2ab + b²)
= (c + d)² - (a - b)²
= (c + d - (a - b))(c + d + (a - b))
= (c + d - a + b)(c + d + a - b)
Answer: (c + d - a + b)(c + d + a - b)

25. a² + 5a + (3 - b)(2 + b)
Notice that (3 - b) + (2 + b) = 5
= a² + [(3 - b) + (2 + b)]a + (3 - b)(2 + b)
= (a + 3 - b)(a + 2 + b)
Answer: (a + 3 - b)(a + 2 + b)

26. x/y + y/x + y/z + z/y + x/z + z/x + 3
= (x/y + x/z + x/x) + (y/x + y/z + y/y) + (z/x + z/y + z/z)
= x(1/y + 1/z + 1/x) + y(1/x + 1/z + 1/y) + z(1/x + 1/y + 1/z)
= (x + y + z)(1/x + 1/y + 1/z)
Answer: (x + y + z)(1/x + 1/y + 1/z)

Case-Study Based Question
Government of India allocated some funds for the refugees who came from Bangladesh for their welfare. The fund is equally distributed among each of the families. The fund allocated is represented by x⁴ + x² + 1 and each of the family received an amount of x² - x + 1.
Based on the above information, answer the following:
(i) Find the number of families received the fund by factorising the given expression.
Number of families = Total Fund / Amount per family
= (x⁴ + x² + 1) / (x² - x + 1)
= (x⁴ + 2x² + 1 - x²) / (x² - x + 1)
= ((x² + 1)² - x²) / (x² - x + 1)
= ((x² - x + 1)(x² + x + 1)) / (x² - x + 1)
= x² + x + 1
Answer: x² + x + 1

(ii) Find the value of x if the number of family was 13.
x² + x + 1 = 13
x² + x - 12 = 0
(x + 4)(x - 3) = 0
Since x should be a positive integer, x = 3.
Answer: 3

(iii) If each of the family received ₹ 1057, then find the positive value of x.
x² - x + 1 = 1057
x² - x - 1056 = 0
(x - 33)(x + 32) = 0
Positive value of x = 33.
Answer: 33