Question

find the G.C.D. and L.C.M. of the polynomial 6xy³ and 8x²y⁴ ( with solutions)​

Answer 1

Let f(x) = 21x2y and g(x) = 35xy2

f(x) = 21x2y = 3 ⋅ 7 ⋅ x2 y

g(x) = 35xy2 = 5 ⋅ 7 ⋅ x y2

L.C.M = 7 ⋅ 3 ⋅ 5 ⋅ x2 ⋅ y2

= 105 x2 y2

GCD = 7 ⋅ x ⋅ y

= 7xy

f(x) × g(x) = LCM × GCD

21x2y (35xy2) = (105 x2 y2)(7xy)

735x3y3 = 735x3y3

So, the relationship verified.

Problem 2 :

Find the LCM and GCD of the following polynomials.

(x3 −1)(x +1) and (x3 +1)

And also verify the relationship that the product of the polynomials is equal to the product of their LCM and GCD.

Solution :

Let f(x) = (x3 −1)(x +1) and g(x) = (x3 +1)

a3 -b3 = (a-b)(a2 + ab + b2)

f(x) = (x3 −1)(x +1) = (x - 1)(x2 + x + 1)(x +1)

g(x) = (x3 +1)

a3 + b3 = (a+b)(a2 - ab + b2)

g(x) = (x3 + 1) = (x + 1)(x2 - x + 1)

L.C.M

= (x-1)(x+1)(x2+x+1)(x2-x+1)

= (x3 - y3)(x3 + y3)

= (x3)2 - (y3)2

= x6 - y6



G.C.D

GCD = x + 1

f(x) × g(x) = LCM × GCD

(x3 −1)(x +1) ⋅ (x3 +1) = (x6 - y6) (x + 1)

(x3)2 - (y3)2(x + 1) = (x6 - y6) (x + 1)

(x6 - y6) (x + 1) = (x6 - y6) (x + 1)

So, the relationship verified.

Problem 3 :

Find the LCM and GCD of the following polynomials.

(x2y + xy2) and (x2 + xy)

And also verify the relationship that the product of the polynomials is equal to the product of their LCM and GCD.

Solution :

Let f(x) = (x2y + xy2) and g(x) = (x2 + xy)

f(x) = (x2y + xy2) = xy (x + y)

g(x) = (x2 + xy) = x(x + y)

L.C.M = xy(x + y)

GCD = x(x + y)

f(x) × g(x) = LCM × GCD

xy (x + y) ⋅ x(x + y) = xy(x + y) ⋅ x(x + y)

x2y(x + y)2 = x2y(x + y)2

So, the relationship verified.