# HOW TO FIND GCD AND LCM OF TWO POLYNOMIALS

## Key Concept

1. To find GCD or LCM of polynomials, first we have to factor the given polynomials.

2. The coefficients of the variables like x as much as possible.

3. If there is quadratic or cubic polynomial, then it has to be factored suitable algebraic identities.

4. To find GCD, multiply the common factors

5. To find LCM, multiply the factors with highest exponents.

## 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 ⋅ x

g(x)  =  35xy2 =  5 ⋅ 7 ⋅ x y

 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. Apart from the stuff given in this section, if you need any other stuff in math, please use our google custom search here.

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