HOW TO FIND GCD AND LCM OF TWO POLYNOMIALS

Key Concept

Practice Problems

Problem 1 :

Find the LCM and GCD of the following polynomials.

21x²y  and  35xy²

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)  =  21x²y and g(x)  =  35xy²

f(x)  =  21x²y  =  3 ⋅ 7 ⋅ x² y 

g(x)  =  35xy² =  5 ⋅ 7 ⋅ x y² 

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)

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)

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.

Kindly mail your feedback to v4formath@gmail.com

We always appreciate your feedback.