Chapter 2 : HCF and LCM

Exercise - 4

1. Find the HCF of the following:

  1. 2x4 - 2y4 and 3x3 + 6x2y - 3xy2 - 6y3
  2. 12(x3 + x2 + x + 1) and 18(x4 - 1)
  3. x3 + 2x2 - 3x and 2x3 + 5x2 - 3x
  4. 2(x4 - y4) and 3(x3 + 2x2y - xy2 - 2y3)
  5. 18(x3 - x2 + x - 1) and 12(x4 - 1)
  6. 45(x4 - x3 - x2) and 75(8x5 + x2)
  7. 36(3x4 + 5x3 - 2x2) and 54(27x4 - x)
  8. 42(2x3 - 5x2 - 3x) and 60(8x4 + x)
  9. 4(x4 - 1) and 6(x3 - x2 - x + 1)

2. Find the LCM of the following polynomials:

  1. 35(x4 - 27x) and 40(2x3 - 5x2 - 3x)
  2. 20(2x3 + 3x2 - 2x) and 45(x4 + 8x)
  3. 15(4x3 - 4x2 + x) and 35(2x2 - 7x + 3)
  4. 25(x2 + 7x + 12) and 15x(x2 - 16)
  5. x(8x3 + 27) and 2x2 (2x2 + 9x + 9)

3. Find the GCD and LCM of the polynomials P(x) and Q(x), where

P(x) = (x3 - 27) (x2 - 3x + 2) and

Q(x) =( x2 + 3x + 9) (x2 - 5x + 6)

4. The LCM and GCD of two polynomials, P(x) and Q(x) are 56(x4 + x) and 4(x2 - x + 1) respectively. If P(x) = 28(x3 + 1), find Q(x) .

5. For what value of k, the g.c.d. of x2 + x - (2k + 2) and 2x2 + kx - 12 is x + 4?

6. Find the value of K for which the g.c.d. of x2 - 2x - 24 and x2 - kx - 6 is x - 6.

7. (x2 - x - 6) is the GCD of the expression (x + 2) (2x2 + ax + 3) and (x - 3) (3x2 + bx + 8). Find the value of a and b.

8. (x + 1) ( x - 4) is the g.c.d. of the polynomials ( x - 4) (2x2 + x - a) and ( x + 1) (2x2 + bx - 12) find a and b .

9. (x - 3) is the g.c.d. of (x3 -2x2 + px + 6) and ( x2 - 5x + q) . Find (6p + 5q) .

10. Find the value of a and b so that the polynomials P(x) and Q(x) have (x - 1) (x + 4) as their HCF:

P(x) = (x2 - 3x + 2) (x2 + 7x + a)

Q(x) = (x2 + 5x + 4) (x2 - 5x + b)

11. Find the value of a and b so that the polynomial x3 + ax2 + bx + 15 is divisible by x2 + 2x - 15.

12. Find the value of p and q so that the polynomial f(x) = px3 + 2x2 - 19x + 9 is divisible by x2 + x - 6.

13. Determine the value of k such that x + 3 is a factor of the polynomial

f(x) = kx3 + x2 - 22x - 21

14. If x - 2 is a factor of x2 + ax - 6 = 0 and x2 - 9x + b = 0, find the value of a and b.

Answers

1. (i) x2 - y2                          (ii) 6(x + 1) (x2 + 1)
    (iii) x(x + 3)                      (iv) x2 - y2
    (v) 6(x - 1) (x2 + 1)         (vi) 15x2(2x + 1)
    (vii) 18x(3x - 1)               (viii) 6x(2x + 1)
    (ix) 2(x2 - 1)

3. G. C. D. = (x3 - 27) (x - 2)
     LCM = (x - 1) (x - 2) (x - 3) (x2 + 3x + 9)

5. K = 5

7. a = -7, b = 10

9. 0

11. a = 1, b = -17

13. K = 2

2. (i) 280x(x - 3) (2x + 1) (x2 + 3x + 9)
    (ii) 180x(x + 2) (2x - 1) (x2 - 2x + 4)
    (iii) 105x((x - 3) (2x - 1)2
    (iv) 75x(x2 - 16) (x + 3)
    (v) 2x2(8x2 + 27) (x + 3)

4. 8(x3 - x2 + x)

6. K = 5

8. a = 1, b= -5

10. a = 12, b = 4

12. p = 3, q = 6

14. a = 1, b = 14

Privious Next
Subjects Maths (Part-1) by Mr. M. P. Keshari
Chapter 1 Linear Equations in Two Variables
Chapter 2 HCF and LCM
Chapter 3 Rational Expression
Chapter 4 Quadratic Equations
Chapter 5 Arithmetic Progressions
Chapter 6 Instalments
Chapter 7 Income Tax
Chapter 8 Similar Triangles