FACTORISATION - Q&A

EXERCISE 13(A)

1. Multiple Choice Type:

Choose the correct answer from the options given below.
(i) -7x² - 14y is equal to:
(a) -21x²y
(b) 14x²y
(c) 7(-x² - 2y)
(d) -7(x² + 2y)

Solution:
Given expression: -7x² - 14y
Taking out -7 as the common factor:
= -7(x² + 2y)
Ans. (d) -7(x² + 2y)

(ii) a(x - y) - b(x - y)² is equal to:
(a) (x - y)(a - by + bx)
(b) (y - x)(a - bx + by)
(c) (x - y)(a - bx - by)
(d) (x - y)(a - bx + by)

Solution:
Given expression: a(x - y) - b(x - y)²
Taking (x - y) common:
= (x - y)[a - b(x - y)]
= (x - y)(a - bx + by)
Ans. (d) (x - y)(a - bx + by)

(iii) a² + bc + ab + ac is equal to:
(a) (a + b)(a + c)
(b) (a + b)(b + c)
(c) (a + c)(a - b)
(d) (a - b)(b + c)

Solution:
Rearranging terms: a² + ab + ac + bc
= a(a + b) + c(a + b)
= (a + b)(a + c)
Ans. (a) (a + b)(a + c)

(iv) 1 - 2x - 2x² + 4x³ is equal to:
(a) (1 + 2x)(1 - 2x²)
(b) (1 + 2x)(1 + 2x²)
(c) (1 - 2x)(1 - 2x²)
(d) (1 - 2x)(1 + 2x²)

Solution:
= 1(1 - 2x) - 2x²(1 - 2x)
= (1 - 2x)(1 - 2x²)
Ans. (c) (1 - 2x)(1 - 2x²)

(v) a(x - y) - b(y - x)² is equal to:
(a) (x - y)(a - by + bx)
(b) (y - x)(a - bx + by)
(c) (x - y)(a - bx - by)
(d) (x - y)(a - bx + by)

Solution:
Note that (y - x)² = [-(x - y)]² = (x - y)²
So, expression becomes: a(x - y) - b(x - y)²
= (x - y)[a - b(x - y)]
= (x - y)(a - bx + by)
Ans. (a) (x - y)(a - by + bx) (Note: bx and by positions swapped in option text but mathematically equivalent).

2. 7a⁶b⁸ - 34a⁴b⁶ + 51a²b⁴

Solution:
H.C.F of 7, 34, 51 is 17 (Wait, 7 is not divisible by 17. 34=2×17, 51=3×17. There is no common numerical factor other than 1. Wait, scanning error possible. If the first term is 17, then 17 is common. But image says 7. Let's assume numerical HCF is 1.)
H.C.F of a⁶, a⁴, a² is a².
H.C.F of b⁸, b⁶, b⁴ is b⁴.
= a²b⁴(7a⁴b⁴ - 34a²b² + 51)
(Note: If the first number was 17, answer would be 17a²b⁴(a⁴b⁴ - 2a²b² + 3))

3. 3x⁵y - 27x⁴y² + 12x³y³

Solution:
H.C.F of 3, 27, 12 is 3.
Variables common: x³y
= 3x³y(x² - 9xy + 4y²)

4. x²(a - b) - y²(a - b) + z²(a - b)

Solution:
Taking (a - b) common:
= (a - b)(x² - y² + z²)

5. (x + y)(a + b) + (x - y)(a + b)

Solution:
Taking (a + b) common:
= (a + b)[(x + y) + (x - y)]
= (a + b)(x + y + x - y)
= (a + b)(2x) = 2x(a + b)

6. 2b(2a + b) - 3c(2a + b)

Solution:
Taking (2a + b) common:
= (2a + b)(2b - 3c)

7. 12abc - 6a²b²c² + 3a³b³c³

Solution:
Common factor: 3abc
= 3abc(4 - 2abc + a²b²c²)

8. 4x(3x - 2y) - 2y(3x - 2y)

Solution:
Taking (3x - 2y) common:
= (3x - 2y)(4x - 2y)
Taking 2 common from second bracket:
= (3x - 2y)2(2x - y)
= 2(3x - 2y)(2x - y)

9. (a + 2b)(3a + b) - (a + b)(a + 2b) + (a + 2b)²

Solution:
Taking (a + 2b) common from all terms:
= (a + 2b)[(3a + b) - (a + b) + (a + 2b)]
= (a + 2b)(3a + b - a - b + a + 2b)
= (a + 2b)(3a + 2b)

10. 6xy(a² + b²) + 8yz(a² + b²) - 10xz(a² + b²)

Solution:
Taking 2(a² + b²) common:
= 2(a² + b²)(3xy + 4yz - 5xz)

11. xy - ay - ax + a² + bx - ab

Solution:
Grouping terms: (xy - ay) - (ax - a²) + (bx - ab)
= y(x - a) - a(x - a) + b(x - a)
= (x - a)(y - a + b)

12. 3x⁵ - 6x⁴ - 2x³ + 4x² + x - 2

Solution:
Grouping: (3x⁵ - 6x⁴) - (2x³ - 4x²) + (x - 2)
= 3x⁴(x - 2) - 2x²(x - 2) + 1(x - 2)
= (x - 2)(3x⁴ - 2x² + 1)

13. -x²y - x + 3xy + 3

Solution:
Grouping: (-x²y - x) + (3xy + 3)
= -x(xy + 1) + 3(xy + 1)
= (xy + 1)(3 - x)

14. 6a² - 3a²b - bc² + 2c²

Solution:
Grouping: (6a² - 3a²b) + (2c² - bc²)
= 3a²(2 - b) + c²(2 - b)
= (2 - b)(3a² + c²)

15. 3a²b - 12a² - 9b + 36

Solution:
Grouping: (3a²b - 12a²) - (9b - 36)
= 3a²(b - 4) - 9(b - 4)
= (b - 4)(3a² - 9)
= (b - 4)3(a² - 3) = 3(b - 4)(a² - 3)

16. x² - (a - 3)x - 3a

Solution:
Opening bracket: x² - ax + 3x - 3a
= x(x - a) + 3(x - a)
= (x - a)(x + 3)

17. ab² - (a - c)b - c

Solution:
Opening bracket: ab² - ab + bc - c
= ab(b - 1) + c(b - 1)
= (b - 1)(ab + c)

18. (a² - b²)c + (b² - c²)a

Solution:
Opening brackets: a²c - b²c + ab² - ac²
Rearranging: a²c - ac² + ab² - b²c
= ac(a - c) + b²(a - c)
= (a - c)(ac + b²)

19. a³ - a² - ab + a + b - 1

Solution:
Rearranging: (a³ - a²) - (ab - b) + (a - 1)
= a²(a - 1) - b(a - 1) + 1(a - 1)
= (a - 1)(a² - b + 1)

20. ab(c² + d²) - a²cd - b²cd

Solution:
Opening bracket: abc² + abd² - a²cd - b²cd
Rearranging: (abc² - a²cd) + (abd² - b²cd)
= ac(bc - ad) - bd(bc - ad)
Note: abd² - b²cd = bd(ad - bc) = -bd(bc - ad)
= (bc - ad)(ac - bd)

21. 2ab² - aby + 2cby - cy²

Solution:
Grouping: (2ab² - aby) + (2cby - cy²)
= ab(2b - y) + cy(2b - y)
= (2b - y)(ab + cy)

22. ax + 2bx + 3cx - 3a - 6b - 9c

Solution:
Grouping: (ax + 2bx + 3cx) - (3a + 6b + 9c)
= x(a + 2b + 3c) - 3(a + 2b + 3c)
= (a + 2b + 3c)(x - 3)

23. 2ab²c - 2a + 3b³c - 3b - 4b²c² + 4c

Solution:
Grouping: (2ab²c - 2a) + (3b³c - 3b) - (4b²c² - 4c)
= 2a(b²c - 1) + 3b(b²c - 1) - 4c(b²c - 1)
= (b²c - 1)(2a + 3b - 4c)

EXERCISE 13(B)

1. Multiple Choice Type:
Choose the correct answer from the options given below.
(i) (2x + y)² - (2y + x)² is equal to:
(a) 3(x + y)(x - y)
(b) 2(x - y)(x + y)
(c) 2(y - x)(x + y)
(d) (3x + y)(3x - y)

Solution:
Using a² - b² = (a + b)(a - b)
= [(2x + y) + (2y + x)][(2x + y) - (2y + x)]
= (3x + 3y)(2x + y - 2y - x)
= 3(x + y)(x - y)
Ans. (a) 3(x + y)(x - y)

(ii) 49 - (x + 5)² is equal to:
(a) (54 - x)(54 + x)
(b) (2 - x)(12 + x)
(c) 48(x + 5)²
(d) 48(x - 5)²

Solution:
= 7² - (x + 5)²
= [7 + (x + 5)][7 - (x + 5)]
= (7 + x + 5)(7 - x - 5)
= (12 + x)(2 - x)
Ans. (b) (2 - x)(12 + x)

(iii) a² - 2ab + b² + a - b is equal to:
(a) (a - b)(a + b - 1)
(b) (a - b)(a + b + 1)
(c) (a + b)(a - b - 1)
(d) (a - b)(a - b + 1)

Solution:
= (a - b)² + (a - b)
= (a - b)(a - b + 1)
Ans. (d) (a - b)(a - b + 1)

(iv) x² + y² - 2xy - 1 is equal to:
(a) (x + y - 1)(x - y - 1)
(b) (x + y + 1)(x - y - 1)
(c) (x + y + 1)(x - y + 1)
(d) (x - y + 1)(x - y - 1)

Solution:
= (x - y)² - 1²
= (x - y + 1)(x - y - 1)
Ans. (d) (x - y + 1)(x - y - 1)

(v) a² + 2a + 1 - b² - x² + 2bx is equal to:
(a) (a + 1 - b + x)(a - 1 - b + x)
(b) (a + 1 + b - x)(a + 1 - b + x)
(c) (a - 1 + b - x)(a - 1 - b + x)
(d) (a - 1 + bx)(a + 1 - bx)

Solution:
= (a² + 2a + 1) - (b² + x² - 2bx)
= (a + 1)² - (b - x)²
= [(a + 1) + (b - x)][(a + 1) - (b - x)]
= (a + 1 + b - x)(a + 1 - b + x)
Ans. (b) (a + 1 + b - x)(a + 1 - b + x)

2. (a + 2b)² - a²

Solution:
= (a + 2b + a)(a + 2b - a)
= (2a + 2b)(2b)
= 2(a + b)(2b) = 4b(a + b)

3. (5a - 3b)² - 16b²

Solution:
= (5a - 3b)² - (4b)²
= (5a - 3b + 4b)(5a - 3b - 4b)
= (5a + b)(5a - 7b)

4. a⁴ - (a² - 3b²)²

Solution:
= (a²)² - (a² - 3b²)²
= [a² + (a² - 3b²)][a² - (a² - 3b²)]
= (2a² - 3b²)(a² - a² + 3b²)
= 3b²(2a² - 3b²)

5. (5a - 2b)² - (2a - b)²

Solution:
= [(5a - 2b) + (2a - b)][(5a - 2b) - (2a - b)]
= (7a - 3b)(3a - b)

6. 1 - 25(a + b)²

Solution:
= 1² - [5(a + b)]²
= [1 + 5(a + b)][1 - 5(a + b)]
= (1 + 5a + 5b)(1 - 5a - 5b)

7. 4(2a + b)² - (a - b)²

Solution:
= [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)
= 3(5a + b)(a + b)

8. 25(2x + y)² - 16(x - y)²

Solution:
= [5(2x + y)]² - [4(x - y)]²
= [5(2x + y) + 4(x - y)][5(2x + y) - 4(x - y)]
= (10x + 5y + 4x - 4y)(10x + 5y - 4x + 4y)
= (14x + y)(6x + 9y)
= 3(14x + y)(2x + 3y)

9. (6 2/3)² - (2 1/3)²

Solution:
= (20/3)² - (7/3)²
= (20/3 + 7/3)(20/3 - 7/3)
= (27/3)(13/3) = 9 × 13/3 = 3 × 13 = 39

10. (0.7)² - (0.3)²

Solution:
= (0.7 + 0.3)(0.7 - 0.3)
= (1.0)(0.4) = 0.4

11. 75(x + y)² - 48(x - y)²

Solution:
Taking 3 common:
= 3[25(x + y)² - 16(x - y)²]
= 3[{5(x + y)}² - {4(x - y)}²]
= 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)

12. a² + 4a + 4 - b²

Solution:
= (a + 2)² - b²
= (a + 2 + b)(a + 2 - b)

13. a² - b² - 2b - 1

Solution:
= a² - (b² + 2b + 1)
= a² - (b + 1)²
= (a + b + 1)(a - b - 1)

14. x² + 6x + 9 - 4y²

Solution:
= (x + 3)² - (2y)²
= (x + 3 + 2y)(x + 3 - 2y)

EXERCISE 13(C)

1. Multiple Choice Type:
Choose the correct answer from the options given below.
(i) x² - 9x - 10 is equal to:
(a) (x - 10)(x + 1)
(b) (x - 10)(x - 1)
(c) (x + 10)(x - 1)
(d) (x + 10)(x + 1)

Solution:
Factors of -10 whose sum is -9 are -10 and 1.
= x² - 10x + x - 10
= x(x - 10) + 1(x - 10)
= (x - 10)(x + 1)
Ans. (a) (x - 10)(x + 1)

(ii) x² - 23x + 42 is equal to:
(a) (x - 21)(x + 2)
(b) (x - 21)(x - 2)
(c) (x + 21)(x + 2)
(d) (x + 21)(x - 2)

Solution:
Factors of 42 whose sum is -23 are -21 and -2.
= (x - 21)(x - 2)
Ans. (b) (x - 21)(x - 2)

(iii) (4x² - 4x + 1) ÷ (2x - 1) is equal to:
(a) 2x + 1
(b) 2x - 1
(c) 2x - 1
(d) none of these

Solution:
Numerator = (2x - 1)²
= (2x - 1)² / (2x - 1)
= 2x - 1
Ans. (b) 2x - 1

(iv) (x + y)² - 3(x + y) - 4 is equal to:
(a) (x + y + 4)(x + y - 1)
(b) (x + y + 4)(x + y + 1)
(c) (x + y - 4)(x + y + 1)
(d) (x + y - 4)(x + y - 1)

Solution:
Let a = x + y. Expression: a² - 3a - 4
Factors of -4 summing to -3 are -4 and 1.
= (a - 4)(a + 1)
= (x + y - 4)(x + y + 1)
Ans. (c) (x + y - 4)(x + y + 1)

(v) 60 + 11x - x² is equal to:
(a) (4 + x)(15 - x)
(b) (4 - x)(15 - x)
(c) (4 + x)(15 - x)
(d) (4 + x)(15 + x)

Solution:
= -(x² - 11x - 60)
Factors of -60 summing to -11 are -15 and 4.
= -(x - 15)(x + 4)
= (15 - x)(4 + x)
Ans. (a) (4 + x)(15 - x)

2. a² + 5a + 6

Solution:
Factors of 6 summing to 5: 3, 2.
= (a + 3)(a + 2)

3. a² - 5a + 6

Solution:
Factors of 6 summing to -5: -3, -2.
= (a - 3)(a - 2)

4. a² + 5a - 6

Solution:
Factors of -6 summing to 5: 6, -1.
= (a + 6)(a - 1)

5. x² + 5xy + 4y²

Solution:
Factors of 4 summing to 5: 4, 1.
= x² + 4xy + xy + 4y²
= x(x + 4y) + y(x + 4y)
= (x + 4y)(x + y)

6. a² - 3a - 40

Solution:
Factors of -40 summing to -3: -8, 5.
= (a - 8)(a + 5)

7. x² - x - 72

Solution:
Factors of -72 summing to -1: -9, 8.
= (x - 9)(x + 8)

8. 3a² - 5a + 2

Solution:
Product = 3 × 2 = 6. Sum = -5. Factors: -3, -2.
= 3a² - 3a - 2a + 2
= 3a(a - 1) - 2(a - 1)
= (a - 1)(3a - 2)

9. 2a² - 17ab + 26b²

Solution:
Product = 2 × 26 = 52. Sum = -17. Factors: -4, -13.
= 2a² - 4ab - 13ab + 26b²
= 2a(a - 2b) - 13b(a - 2b)
= (a - 2b)(2a - 13b)

10. 2x² + xy - 6y²

Solution:
Product = -12. Sum = 1. Factors: 4, -3.
= 2x² + 4xy - 3xy - 6y²
= 2x(x + 2y) - 3y(x + 2y)
= (x + 2y)(2x - 3y)

11. 4c² + 3c - 10

Solution:
Product = -40. Sum = 3. Factors: 8, -5.
= 4c² + 8c - 5c - 10
= 4c(c + 2) - 5(c + 2)
= (c + 2)(4c - 5)

12. 14x² + x - 3

Solution:
Product = -42. Sum = 1. Factors: 7, -6.
= 14x² + 7x - 6x - 3
= 7x(2x + 1) - 3(2x + 1)
= (2x + 1)(7x - 3)

13. 6 + 7b - 3b²

Solution:
Product = -18. Sum = 7. Factors: 9, -2.
= 6 + 9b - 2b - 3b²
= 3(2 + 3b) - b(2 + 3b)
= (2 + 3b)(3 - b)

14. 5 + 7x - 6x²

Solution:
Product = -30. Sum = 7. Factors: 10, -3.
= 5 + 10x - 3x - 6x²
= 5(1 + 2x) - 3x(1 + 2x)
= (1 + 2x)(5 - 3x)

15. 4 + y - 14y²

Solution:
Product = -56. Sum = 1. Factors: 8, -7.
= 4 + 8y - 7y - 14y²
= 4(1 + 2y) - 7y(1 + 2y)
= (1 + 2y)(4 - 7y)

16. 5 + 3a - 14a²

Solution:
Product = -70. Sum = 3. Factors: 10, -7.
= 5 + 10a - 7a - 14a²
= 5(1 + 2a) - 7a(1 + 2a)
= (1 + 2a)(5 - 7a)

17. (2a + b)² + 5(2a + b) + 6

Solution:
Let x = 2a + b. Expression: x² + 5x + 6
= (x + 2)(x + 3)
= (2a + b + 2)(2a + b + 3)

18. 1 - (2x + 3y) - 6(2x + 3y)²

Solution:
Let u = 2x + 3y. Expression: 1 - u - 6u²
Product = -6. Sum = -1. Factors: -3, 2.
= 1 - 3u + 2u - 6u²
= 1(1 - 3u) + 2u(1 - 3u)
= (1 - 3u)(1 + 2u)
= [1 - 3(2x + 3y)][1 + 2(2x + 3y)]
= (1 - 6x - 9y)(1 + 4x + 6y)

19. (x - 2y)² - 12(x - 2y) + 32

Solution:
Let u = x - 2y. Expression: u² - 12u + 32
Factors of 32 summing to -12: -8, -4.
= (u - 8)(u - 4)
= (x - 2y - 8)(x - 2y - 4)

20. 8 + 6(a + b) - 5(a + b)²

Solution:
Let x = a + b. Expression: 8 + 6x - 5x²
Product = -40. Sum = 6. Factors: 10, -4.
= 8 + 10x - 4x - 5x²
= 2(4 + 5x) - x(4 + 5x)
= (4 + 5x)(2 - x)
= (4 + 5a + 5b)(2 - a - b)

21. 2(x + 2y)² - 5(x + 2y) + 2

Solution:
Let u = x + 2y. Expression: 2u² - 5u + 2
Product = 4. Sum = -5. Factors: -4, -1.
= 2u² - 4u - u + 2
= 2u(u - 2) - 1(u - 2)
= (u - 2)(2u - 1)
= (x + 2y - 2)(2(x + 2y) - 1)
= (x + 2y - 2)(2x + 4y - 1)

22. In each case, find whether the trinomial is a perfect square or not:
(i) x² + 14x + 49

Solution:
x² + 2(7)x + 7² = (x + 7)². Yes.

(ii) a² - 10a + 25

Solution:
a² - 2(5)a + 5² = (a - 5)². Yes.

(iii) 4x² + 4x + 1

Solution:
(2x)² + 2(2x)(1) + 1² = (2x + 1)². Yes.

(iv) 9b² + 12b + 16

Solution:
(3b)² + 12b + 4². Middle term for perfect square should be 2(3b)(4) = 24b. Here it is 12b. No.

(v) 16x² - 16xy + y²

Solution:
(4x)² - 16xy + y². Middle term should be 2(4x)(y) = 8xy. Here it is 16xy. No.

(vi) x² - 4x + 16

Solution:
x² - 4x + 4². Middle term should be 2(x)(4) = 8x. Here it is 4x. No.

EXERCISE 13(D)

1. Multiple Choice Type:
Choose the correct answer from the options given below.
(i) x³ - 4x is equal to:
(a) x(x + 4)(x - 4)
(b) x(x + 2)(x - 2)
(c) (x + 4)(x - 4)
(d) (x + 2)(x - 2)

Solution:
= x(x² - 4)
= x(x² - 2²)
= x(x + 2)(x - 2)
Ans. (b) x(x + 2)(x - 2)

(ii) x⁴ - y⁴ + x² - y² is equal to:
(a) (x + y + 1)(x + y - 1)(x² + y²)
(b) (x + y)(x - y)(x² + y² - 1)
(c) (x + y)(x - y)(x² + y² + 1)
(d) none of these

Solution:
= (x² - y²)(x² + y²) + (x² - y²)
= (x² - y²)(x² + y² + 1)
= (x + y)(x - y)(x² + y² + 1)
Ans. (c) (x + y)(x - y)(x² + y² + 1)

(iii) x³ - x² + ax + x - a - 1 is equal to:
(a) (x - 1)(x² + a - 1)
(b) (x - 1)(x² + a + 1)
(c) (x - 1)(x² - a + 1)
(d) (x - 1)(x² - a - 1)

Solution:
Rearranging: x³ - x² + x - 1 + ax - a
= x²(x - 1) + 1(x - 1) + a(x - 1)
= (x - 1)(x² + 1 + a)
Ans. (b) (x - 1)(x² + a + 1)

(iv) 8x³ - 18x is equal to:
(a) x(2x + 3)(2x - 3)
(b) 2x(3 - 2x)(3 + 2x)
(c) 2x(2x + 3)(2x - 3)
(d) x(4x + 6y)(4x - 6y)

Solution:
= 2x(4x² - 9)
= 2x((2x)² - 3²)
= 2x(2x + 3)(2x - 3)
Ans. (c) 2x(2x + 3)(2x - 3)

(v) x² - (a - b)x - ab is equal to:
(a) (x - a)(x - b)
(b) (x + a)(x - b)
(c) (x - a)(x + b)
(d) (x + a)(x + b)

Solution:
= x² - ax + bx - ab
= x(x - a) + b(x - a)
= (x - a)(x + b)
Ans. (c) (x - a)(x + b)

2. 8x²y - 18y³

Solution:
= 2y(4x² - 9y²)
= 2y((2x)² - (3y)²)
= 2y(2x + 3y)(2x - 3y)

3. 25x³ - x

Solution:
= x(25x² - 1)
= x((5x)² - 1²)
= x(5x + 1)(5x - 1)

4. 16x⁴ - 81y⁴

Solution:
= (4x²)² - (9y²)²
= (4x² + 9y²)(4x² - 9y²)
= (4x² + 9y²)((2x)² - (3y)²)
= (4x² + 9y²)(2x + 3y)(2x - 3y)

5. x² - y² - 3x - 3y

Solution:
= (x + y)(x - y) - 3(x + y)
= (x + y)(x - y - 3)

6. x² - y² - 2x + 2y

Solution:
= (x + y)(x - y) - 2(x - y)
= (x - y)(x + y - 2)

7. 3x² + 15x - 72

Solution:
= 3(x² + 5x - 24)
Factors of -24 summing to 5: 8, -3.
= 3(x + 8)(x - 3)

8. 2a² - 8a - 64

Solution:
= 2(a² - 4a - 32)
Factors of -32 summing to -4: -8, 4.
= 2(a - 8)(a + 4)

9. 3x²y + 11xy + 6y

Solution:
= y(3x² + 11x + 6)
Product 18, Sum 11. Factors: 9, 2.
= y(3x² + 9x + 2x + 6)
= y[3x(x + 3) + 2(x + 3)]
= y(3x + 2)(x + 3)

10. 5ap² + 11ap + 2a

Solution:
= a(5p² + 11p + 2)
Product 10, Sum 11. Factors: 10, 1.
= a(5p² + 10p + p + 2)
= a[5p(p + 2) + 1(p + 2)]
= a(5p + 1)(p + 2)

11. a² + 2ab + b² - c²

Solution:
= (a + b)² - c²
= (a + b + c)(a + b - c)

12. x² + 6xy + 9y² + x + 3y

Solution:
= (x + 3y)² + 1(x + 3y)
= (x + 3y)(x + 3y + 1)

13. 4a² - 12ab + 9b² + 4a - 6b

Solution:
= (2a - 3b)² + 2(2a - 3b)
= (2a - 3b)(2a - 3b + 2)

14. 2a²b² - 98b⁴

Solution:
= 2b²(a² - 49b²)
= 2b²(a + 7b)(a - 7b)

15. a² - 16b² - 2a - 8b

Solution:
= (a + 4b)(a - 4b) - 2(a + 4b)
= (a + 4b)(a - 4b - 2)

Test yourself

1. Multiple Choice Type:
Choose the correct answer from the options given below.
(i) (a + b)² - 4ab is equal to:
(a) (a + b + 2ab)(a + b - 2ab)
(b) (a + b)(a - b)
(c) (a + b)(a + b)
(d) (a - b)(a - b)

Solution:
= a² + 2ab + b² - 4ab
= a² - 2ab + b²
= (a - b)²
Ans. (d) (a - b)(a - b)

(ii) a⁴ + 4a² - 32 is equal to :
(a) (a² + 8)(a + 2)(a + 2)
(b) (a² - 8)(a - 2)(a + 2)
(c) (a² + 8)(a² + 4)
(d) (a² + 8)(a + 2)(a - 2)

Solution:
Let x = a². Expression: x² + 4x - 32
Factors of -32 sum to 4: 8, -4.
= (x + 8)(x - 4)
= (a² + 8)(a² - 4)
= (a² + 8)(a + 2)(a - 2)
Ans. (d) (a² + 8)(a + 2)(a - 2)

(iii) -60y + 25y² is equal to:
(a) (3 + 5y)(3 + 5y)
(b) (3 - 5y)(6 - 5y)
(c) (3 + 4y)(3 - 4y)
(d) none of these

Solution:
Note: The question text appears incomplete in the source image (likely "36 - 60y + 25y^2").
However, based on the provided text, -60y + 25y² = 5y(5y - 12). None of the options match this result directly.
If we assume the question is 36 - 60y + 25y², then = (6 - 5y)². None of the options exactly match this either.
Given the options, none seem correct for the fragment provided.
Ans. (d) none of these

(iv) (x - 2y)² - 3x + 6y is equal to:
(a) (x - 3y)(x + 2y)
(b) (x - 2y)(x - 2y + 3)
(c) (x + 2y - 3)(x + 2y)
(d) (x - 2y)(x - 2y - 3)

Solution:
= (x - 2y)² - 3(x - 2y)
= (x - 2y)(x - 2y - 3)
Ans. (d) (x - 2y)(x - 2y - 3)

(v) a(x - y)² - by + bx is equal to :
(a) (x - y)(ax + by + b)
(b) (x - y)(ax + by - b)
(c) (x - y)(x + y + a - b)
(d) (x - y)(ax - ay + b)

Solution:
= a(x - y)² + b(x - y)
= (x - y)[a(x - y) + b]
= (x - y)(ax - ay + b)
Ans. (d) (x - y)(ax - ay + b)

(vi) Statement 1: The product of two binomials is a trinomial, conversely if we factorise a trinomial we always obtain two binomial factors.
Statement 2: The square of the difference of two terms = The sum of the same two terms x their difference.
Which of the following options is correct?
(a) Both the statements are true.
(b) Both the statements are false.
(c) Statement 1 is true, and statement 2 is false.
(d) Statement 1 is false, and statement 2 is true.

Solution:
Statement 1 is false (e.g., product of (x+1)(x-1) = x² - 1, which is a binomial).
Statement 2 is false (This describes a² - b², not (a-b)²).
Ans. (b) Both the statements are false.

The following questions are Assertion-Reason based questions. Choose your answer based on the codes given below.
(1) Both A and R are correct, and R is the correct explanation for A.
(2) Both A and R are correct, and R is not the correct explanation for A.
(3) A is true, but R is false.
(4) A is false, but R is true.

(vii) Assertion (A): 25x² - 5x + 1 is a perfect square trinomial.
Reason (R): Any trinomial which can be expressed as x² + y² + 2xy or x² + y² - 2xy is a perfect square trinomial.

Solution:
Check A: (5x)² - 5x + 1². Middle term should be 2(5x)(1) = 10x. It is 5x. So A is False.
Check R: True (definition of perfect square trinomial).
Ans. (4) A is false, but R is true.

(viii) Assertion (A): x² + 7x + 12 = (x + 4)(x + 3)
Reason (R): To factorise a given trinomial, the product of the first and the last term of the trinomial is always the sum of the two parts when we split the middle term.

Solution:
Check A: True.
Check R: False. The product of the two parts must equal the product of first and last terms. Their sum must equal the middle term.
Ans. (3) A is true, but R is false.

(ix) Assertion (A): The value of k so that the factors of (x² - kx + 121/16) are the same is 11/2.
Reason(R): (x + a)(x + b) = x² + (a + b)x + ab

Solution:
For factors to be the same, it must be a perfect square.
x² - kx + (11/4)² = (x - 11/4)²
k = 2(11/4) = 11/2. A is True.
R is the identity for multiplication, but doesn't explicitly explain the condition for equal factors (perfect square) as directly as the discriminant, though it is the basis. However, since A is true and R is a correct identity, but R is the general formula for factorization, usually "correct explanation" implies the specific condition ($b^2-4ac=0$).
However, looking at the context, this maps to code (2) or (1). Let's assume (1) as it explains the structure of the trinomial.
Ans. (1) Both A and R are correct, and R is the correct explanation for A. (or 2 depending on strictness).

(x) Assertion (A): There are two values of b so that x² + by - 24 is factorisable.
Reason (R): Two values have Product = -24 and sum = 2.

Solution:
A implies b can only have 2 values. If we consider integer factorization, b can be ±2, ±5, ±10, ±23. So A is False.
R is a true statement (6 and -4 have product -24 and sum 2).
Ans. (4) A is false, but R is true.

2. Factorise:
(i) 6x³ - 8x²

Solution:
= 2x²(3x - 4)

(ii) 36x²y² - 30x³y³ + 48x³y²

Solution:
= 6x²y²(6 - 5xy + 8x)

(iii) 8(2a + 3b)³ - 12(2a + 3b)²

Solution:
= 4(2a + 3b)²[2(2a + 3b) - 3]
= 4(2a + 3b)²(4a + 6b - 3)

(iv) 9a(x - 2y)⁴ - 12a(x - 2y)³

Solution:
= 3a(x - 2y)³[3(x - 2y) - 4]
= 3a(x - 2y)³(3x - 6y - 4)

3. Factorise:
(i) a² - ab(1 - b) - b³

Solution:
= a² - ab + ab² - b³
= a(a - b) + b²(a - b)
= (a - b)(a + b²)

(ii) xy² + (x - 1)y - 1

Solution:
= xy² + xy - y - 1
= xy(y + 1) - 1(y + 1)
= (y + 1)(xy - 1)

(iii) (ax + by)² + (bx - ay)²

Solution:
= a²x² + b²y² + 2abxy + b²x² + a²y² - 2abxy
= a²x² + b²x² + b²y² + a²y²
= x²(a² + b²) + y²(b² + a²)
= (a² + b²)(x² + y²)

(iv) ab(x² + y²) - xy(a² + b²)

Solution:
= abx² + aby² - a²xy - b²xy
= abx² - a²xy + aby² - b²xy
= ax(bx - ay) - by(-ay + bx)
= (bx - ay)(ax - by)

(v) m - 1 - (m - 1)² + am - a

Solution:
= (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)

4. Factorise :
(i) 25(2x - y)² - 16(x - 2y)²

Solution:
= [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)
= (14x - 13y)(6x + 3y)
= 3(2x + y)(14x - 13y)

(ii) 16(5x + 4)² - 9(3x - 2)²

Solution:
= [4(5x + 4)]² - [3(3x - 2)]²
= [4(5x + 4) + 3(3x - 2)][4(5x + 4) - 3(3x - 2)]
= (20x + 16 + 9x - 6)(20x + 16 - 9x + 6)
= (29x + 10)(11x + 22)
= 11(29x + 10)(x + 2)

(iii) 25(x - 2y)² - 4

Solution:
= [5(x - 2y)]² - 2²
= [5(x - 2y) + 2][5(x - 2y) - 2]
= (5x - 10y + 2)(5x - 10y - 2)

5. Factorise:
(i) a² - 23a + 42

Solution:
= (a - 21)(a - 2)

(ii) a² - 23a - 108

Solution:
Factors of -108 summing to -23: -27, 4.
= (a - 27)(a + 4)

(iii) 1 - 18x - 63x²

Solution:
Product = -63. Sum = -18. Factors: -21, 3.
= 1 + 3x - 21x - 63x²
= 1(1 + 3x) - 21x(1 + 3x)
= (1 + 3x)(1 - 21x)

(iv) 5x² - 4xy - 12y²

Solution:
Product = -60. Sum = -4. Factors: -10, 6.
= 5x² - 10xy + 6xy - 12y²
= 5x(x - 2y) + 6y(x - 2y)
= (x - 2y)(5x + 6y)

(v) x(3x + 14) + 8

Solution:
= 3x² + 14x + 8
Product = 24. Sum = 14. Factors: 12, 2.
= 3x² + 12x + 2x + 8
= 3x(x + 4) + 2(x + 4)
= (x + 4)(3x + 2)

(vi) 5 - 4x(1 + 3x)

Solution:
= 5 - 4x - 12x²
Product = -60. Sum = -4. Factors: -10, 6.
= 5 - 10x + 6x - 12x²
= 5(1 - 2x) + 6x(1 - 2x)
= (1 - 2x)(5 + 6x)

(vii) x²y² - 3xy - 40

Solution:
Let u = xy. u² - 3u - 40.
= (u - 8)(u + 5)
= (xy - 8)(xy + 5)

(viii) (3x - 2y)² - 5(3x - 2y) - 24

Solution:
Let u = 3x - 2y. u² - 5u - 24.
= (u - 8)(u + 3)
= (3x - 2y - 8)(3x - 2y + 3)

(ix) 12(a + b)² - (a + b) - 35

Solution:
Let x = a + b. 12x² - x - 35.
Product = -420. Sum = -1. Factors: -21, 20.
= 12x² - 21x + 20x - 35
= 3x(4x - 7) + 5(4x - 7)
= (4x - 7)(3x + 5)
= (4(a + b) - 7)(3(a + b) + 5)

6. Factorise:
(i) 15(5x - 4)² - 10(5x - 4)

Solution:
= 5(5x - 4)[3(5x - 4) - 2]
= 5(5x - 4)(15x - 12 - 2)
= 5(5x - 4)(15x - 14)

(ii) 3a²x - bx + 3a² - b

Solution:
= x(3a² - b) + 1(3a² - b)
= (3a² - b)(x + 1)

(iii) b(c - d)² + a(d - c) + 3(c - d)

Solution:
= 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 + b²y - ab² - x²y

Solution:
= (ax² - ab²) - (x²y - b²y)
= a(x² - b²) - y(x² - b²)
= (x² - b²)(a - y)
= (x + b)(x - b)(a - y)

(v) 1 - 3x - 3y - 4(x + y)²

Solution:
= 1 - 3(x + y) - 4(x + y)²
Let u = x + y. 1 - 3u - 4u²
= 1 - 4u + u - 4u²
= 1(1 - 4u) + u(1 - 4u)
= (1 - 4u)(1 + u)
= (1 - 4x - 4y)(1 + x + y)

7. Factorise:
(i) 2a³ - 50a

Solution:
= 2a(a² - 25)
= 2a(a + 5)(a - 5)

(ii) 54a²b² - 6

Solution:
= 6(9a²b² - 1)
= 6[(3ab)² - 1²]
= 6(3ab + 1)(3ab - 1)

(iii) 64a²b - 144b³

Solution:
= 16b(4a² - 9b²)
= 16b[(2a)² - (3b)²]
= 16b(2a + 3b)(2a - 3b)

(iv) (2x - y)³ - (2x - y)

Solution:
= (2x - y)[(2x - y)² - 1]
= (2x - y)(2x - y + 1)(2x - y - 1)

(v) x² - 2xy + y² - z²

Solution:
= (x - y)² - z²
= (x - y + z)(x - y - z)

(vi) x² - y² - 2yz - z²

Solution:
= x² - (y² + 2yz + z²)
= x² - (y + z)²
= (x + y + z)(x - y - z)

(vii) 7a⁵ - 567a

Solution:
= 7a(a⁴ - 81)
= 7a((a²)² - 9²)
= 7a(a² + 9)(a² - 9)
= 7a(a² + 9)(a + 3)(a - 3)

(viii) 5x² - 20x⁴/9

Solution:
= 5x²(1 - 4x²/9)
= 5x²(1² - (2x/3)²)
= 5x²(1 + 2x/3)(1 - 2x/3)

8. Factorise xy² - xz², Hence, find the value of:
(i) 9 × 8² - 9 × 2²

Solution:
Factorisation: xy² - xz² = x(y² - z²) = x(y + z)(y - z)
Value: Here x = 9, y = 8, z = 2
= 9(8 + 2)(8 - 2)
= 9(10)(6) = 540

(ii) 40 × 5.5² - 40 × 4.5²

Solution:
Here x = 40, y = 5.5, z = 4.5
= 40(5.5 + 4.5)(5.5 - 4.5)
= 40(10)(1) = 400

9. Factorise :
(i) (a - 3b)² - 36b²

Solution:
= (a - 3b)² - (6b)²
= (a - 3b + 6b)(a - 3b - 6b)
= (a + 3b)(a - 9b)

(ii) 25(a - 5b)² - 4(a - 3b)²

Solution:
= [5(a - 5b)]² - [2(a - 3b)]²
= [5a - 25b + 2a - 6b][5a - 25b - (2a - 6b)]
= (7a - 31b)(3a - 19b)

(iii) a² - 0.36b²

Solution:
= a² - (0.6b)²
= (a + 0.6b)(a - 0.6b)

(iv) x⁴ - 5x² - 36

Solution:
Let u = x². u² - 5u - 36.
Factors of -36 summing to -5: -9, 4.
= (u - 9)(u + 4)
= (x² - 9)(x² + 4)
= (x + 3)(x - 3)(x² + 4)

(v) 15(2x - y)² - 16(2x - y) - 15

Solution:
Let u = 2x - y. 15u² - 16u - 15.
Product = -225. Sum = -16. Factors: -25, 9.
= 15u² - 25u + 9u - 15
= 5u(3u - 5) + 3(3u - 5)
= (3u - 5)(5u + 3)
= (3(2x - y) - 5)(5(2x - y) + 3)
= (6x - 3y - 5)(10x - 5y + 3)

10. Evaluate (using factors): 301² × 300 - 300³

Solution:
= 300(301² - 300²)
= 300(301 + 300)(301 - 300)
= 300(601)(1)
= 180300

11. Use factor method to evaluate :
(i) (5z² - 80) ÷ (z - 4)

Solution:
Numerator = 5(z² - 16) = 5(z + 4)(z - 4)
Expression = 5(z + 4)(z - 4) / (z - 4)
= 5(z + 4)

(ii) 10y(6y + 21) ÷ (2y + 7)

Solution:
Numerator = 10y × 3(2y + 7) = 30y(2y + 7)
Expression = 30y(2y + 7) / (2y + 7)
= 30y

(iii) (a² - 14a - 32) ÷ (a + 2)

Solution:
Numerator: a² - 14a - 32 = (a - 16)(a + 2)
Expression = (a - 16)(a + 2) / (a + 2)
= a - 16

(iv) 39x³(50x² - 98) ÷ 26x²(5x + 7)

Solution:
Numerator = 39x³ × 2(25x² - 49) = 78x³(5x + 7)(5x - 7)
Expression = [78x³(5x + 7)(5x - 7)] / [26x²(5x + 7)]
= (78/26)x^3-2(5x - 7)
= 3x(5x - 7)