Students and Teacher's Forum

Students and Teachers Forum

1. Find the HCF of:

x³ – 8 and x² + 2x + 4.

Here, First expression = x3 - 8 = x3 - 23 = (x - 2) (x2 + x.2 + .....

2. Find the LCM of:

3x² – 22x + 19 and 2x² + 3x – 5

Here, 1st expression = 3x2 – 22x + 19 = 3x2 – (19 + 3)x + 19 = 3x2 .....

3. Find the LCM of:

a² – b² + 2ab – c², (a + b)² – c² and (a + b – c)²

Here, 1st expression: = a2 – b2 + 2ab – c2 2nd expression: = (a + b)2 .....

4. Find the HCF and LCM of:

1 + 4x + 4x² – 16x⁴ and 1 + 2x – 8x³ – 16x⁴

Here, 1st expression = 1 + 4x + 4x2 – 16x4 = 12 + 2. 1. 2x + (2x)2 – 16x4 .....

5. Find the HCF of:

4x⁴ + 16x³ – 20x², 3x³ + 14x² – 5x and x⁴ + 125x

Here, 1st expression: = 4x4 + 16x3 – 20x2 = 4x2(x2 + 4x – 5) .....

6. Find the HCF and LCM of:

8x³ + y³, 8x³ – y³ and 16x⁴ + 4x²y² + y⁴

Here, 1st expression: = 8x3 + y3 = (2x)3 + y3 .....

7. Find HCF and LCM
x⁴+4 and 3x²-6x+6

Here, Expression I: = x4 + 4 =(x2)2 + 22 =(x2+2)2-2.x2.2 .....

8. Find HCF and LCM:
x²+5x+6 and x²-4

Here, Expression I: = x2+5x+6 = x2+2x+3x+6 = .....

9. Find the HCF of:

8a³x(a³ – x³),
4a²x²(a⁴ – x⁴) and
6ax³(a² + 4ax – 5x²)

Solution: 1st expression = 8a3x(a3 – x3) = 2.2.2. a3x(a – x)(a2 + ax + x2). 2nd expression = 4a2x2(a4 – x4) .....

10.

Find the L.C.M. of:

m⁴ + m²+ 1, m⁴ – m and 2m³ + 2m² + 2m

Here,

.....

11. Find the HCF and LCM of:

3a⁴ – 8a³ + 4a², a³ – 4a and 4a³ – 10a² + 4a

Here, 1st expression = 3a4 – 8a3 + 4a2 = a2(3a2 – 8a + 4) = a2{3a2 .....

12. Find the L.C.M:

12a(ab – b²)², 35(a⁴b² – a²b⁴) and 15b(a² + ab)²

Here, 1st expression = 12a(ab – b2)2 = 2 × 2 × 3a × b2 (a – b)2. 2nd expression = 35(a4b2 – .....

13. Find the H.C.F and L.C.M from the following expression:

4x⁴ + 19x²y² + 49y⁴ and 4x³ + 14xy² – 6x²y

Here, 1st expression = 4x4 + 19x2y2 + 49y4 = (2x2)2 + 19x2y2 + 49y4 = (2x2)2 + 2. .....

14. Find the H.C.F and L.C.M of following expression:

8x³ + y³, 8x³ – y³ and 16x⁴ + 4x²y² + y⁴

Here, 1st expression = 8x3 + y3 = (2x)3 + y3 = (2x + y) (4x2 .....

15. Find the LCM of:

x² + 2xy + y² – z², y² + 2yz + z² – x² and z² + 2zx + x² – y²

Here, 1st expression: = x2 + 2xy + y2 – z2 = (x + y)2 – z2 .....

16. Find the HCF of:

16x⁴ – 4x²– 4x – 1 and 8x³ – 1

Here, First expression = 16x4 – 4x2 – 4x – 1 = 16x4– (4x2+ 4x + 1) .....

17. Find the H.C.F and L.C.M from the following expression:

1 + 4x + 4x² – 16x⁴ and 1 + 2x – 8x³ – 16x⁴

Here, 1st expression = 1 + 4x + 4x2 – 16x4 = 12 + 2. 1. 2x + (2x)2 – 16x4 .....

18. Find the HCF of:

1 + 6a + 9a² – 81a⁴, 1 + 6a + 18a² + 27a³, 1 – 27a³

Here, 1st expression: = 1 + 6a + 9a2 – 81a4 = (1)2 + 2.1.3a + (3a)2 – (9a2)2 .....

19. Find the LCM of:

(a – b)² + 4ab and (a + b)³ – 3ab(a + b)

Here, 1st expression = (a – b)2 + 4ab = a2– 2ab + b2+ 4ab = a2+ 2ab + .....

20. Find the HCF of:

a⁵b³ – a³b⁵ and a⁵b⁴ + a⁴b⁵

Here, First expression = a5b3 – a3b5 = a3b3(a2 – b2) = .....

21. Find the LCM of:

(a + b)² – 4ab and (a – b)³ + 3ab(a – b)

Here, 1st expression = (a + b)2 – 4ab = a2 + 2ab + b2 – 4ab = a2 .....

22. Find the LCM of:

8a³ + b³ and 16a⁴ + 4a²b² + b⁴

Here, 1st expression = 8a3 + b3 = (2a)3 + (b)3 = (2a + b) (4a2– 2ab + .....

23. Find the LCM of:

x³– 3x² – x + 3 and x³ – x² – 9x + 9

Here, 1st expression = x3 – 3x2 – x + 3 = x2(x – 3) – 1(x – 3) .....

24. Find the HCF of:

6x² – x – 1 and 54x⁴ + 2x

Here, First expression = 6x2 – x – 1 = 6x2 – 3x + 2x – 1 .....

25. Find the HCF of:

16a⁴ – 4a² – 4a – 1, 16a⁴ + 8a³ – 2a – 1, 8a³ – 1

Here, 1st expression = 16a4 – 4a2 – 4a – 1 = 16a4 – {(2a)2 + 2.2a.1 + (1)2} .....

26. Find the HCF of:

8x³ + 64 and 4x² + 16x + 16

Here, First expression : = 8x3 + 64 = 8(x3 + 8) .....

27. Find the HCF of:

m³ – n³, m⁴ + m²n² + n⁴and m³ + m²n + mn²

Here, 1st expression : = m3 – n3 = (m – n)(m2 + mn + n2). 2nd expression: .....

28. Find the HCF and LCM of:

1 + 4x + 4x² – 16x⁴ and 1 + 2x – 8x³ – 16x⁴

Here, 1st expression = 1 + 4x + 4x2 – 16x4 = 12 + 2. 1. 2x + (2x)2 – .....

29. Find HCF and LCM:
x³+27 and x²-9

Here, Expression I: = x3+27 =x3+33 =(x+3) (x2-3x+9). Expression II: .....

30. Find HCF and LCM:
x²-9 and 3x+9

Here, Expression I: =x2-9 =x2-32 =(x-3) .....

31. Find the LCM:

p⁴ + 6p³ + 8p², p³ + 8p²+ 16p and p⁵ – 16p³

Here, 1st expression = p4 + 6p3 + 8p2 =p2(p2 +6p +8) = p2{p2 + (4 + 2)p + .....

32.

Find the H.C.F. of:

x² + 3x + 2 and x² + 5x + 6

Solution:

.....

33. Find the HCF of:

9x² – 4y² – 8yz – 4z²,
4z² – 4y² – 9x² – 12xy and
9x² + 12xz + 4z² – 4y²

Here, 1st expression = 9x2 – 4y2 – 8yz – 4z2 = 9x2 – (4y2 + 8yz + 4z2) .....

34. Find the HCF and LCM of:

16 (a² + 2a – 3), 24(a² + a – 6) and 48(a² – a – 12)

Here, 1st expression = 16 (a2 + 2a – 3) = 16{a2 + (3 – 1)a – 3} = .....

35.

Find the LCM of:

6x² – x – 1 and 54x⁴ + 2x

Here, 1st expression = 6x2 – x – 1 = 6x2 – (3 – 2)x – 1 .....

36.

Find the H.C.F. of:

3x² – 8x + 4, 2x² – 5x + 2, x⁴ – 8x

Here,

.....

37. Find the HCF and LCM of:

8x³ + y³, 8x³ – y³ and 16x⁴ + 4x²y² + y⁴

Solution: 1st expression = 8x3 + y3 = (2x)3 + y3 .....

38. Find the L.C.M. of:

x² – 9 and (x – 3)²

Here, 1st expression = x2 – 9 = x2 – 32 .....

39. Find the LCM of:

x³– 2x² – 3x, x³ – 9x and x⁴ – 27x

Here, 1st expression = x3 – 2x2 – 3x = x(x2 – 2x – 3) = .....

40. Find the LCM of:

p⁴+ 6p³ + 8p², p³ + 8p²+ 16p and p⁵ – 16p³

Here, 1st expression = p4 + 6p3 + 8p2 =p2(p2 +6p +8) = p2{p2 + (4 + 2)p + 8} .....

41. Find the LCM of:

(x² + 4xy – 21y²), 2x(x² – 6xy + 9y²) and (x³ + 343y³)

Here, 1st expression = (x2 + 4xy – 21y2) = x2 + (7 – 3)xy – 21y = .....

42. Find the LCM of:

x³– 2x² – 3x, x³ – 9x and x4 – 27x

Here, 1st expression: = x3 – 2x2 – 3x = x(x2 – 2x – .....

43. Find the LCM of:

6x² – x – 1 and 54x⁴ + 2x

Here, 1st expression = 6x2 – x – 1 = 6x2 – (3 – 2)x – 1 = 6x2 .....

44. Find LCM of;
x³ -9x, x⁴ – 2x³ -3x²& x³ -27

Here 1st expression: x3 -9x = x( x2 – 32) = x(x+3)(x-3) 2nd expression: x4 – .....

45. Find the HCF of:

x³ – 8 and x² + 2x + 4

Here, First expression = x3 - 8 = x3 - 23 = (x - 2) (x2 + x.2 + 22) .....

46. If x and y are both prime numbers. Find LCM and HCF of x and y.

Here, Since, x and y are prime numbers. LCM = xy HCF = 1 .....

47. Find the HCF of:

(x -1)² and x² – 3x + 2

Here, First expression = (x – 1)2 = (x - 1) (x -1). Second expression = x2 - 3x + 2 .....

48. Find the H.C.F and L.C.M of following expression:

x³ – 3x + 2, x² – 4x + 3 and x² – 5x + 6

Here, 1st expression = x3 – 3x + 2 = x3 – x2 + x2 – x – 2x + 2 .....

49. The L.C.M. of two prime numbers is equal to

product of two .....

50. Find the LCM of:

x⁴ + x²y² + y⁴ and x³ – y³

Here, 1st expression, = x4 + x2y2 + y4 = (x2)2 + (y2)2 + x2y2 .....