# HOW TO FIND GCD AND LCM OF TWO POLYNOMIALS

How to Find GCD and LCM of Two Polynomials :

To find GCXD or LCM, first we have to find the factors of the given expression or polynomial.

• If we have coefficients of x or y term, then we have to decompose the coefficients as much as possible.
• If we have quadratic or cubic expression, then we have to factorize using suitable algebraic identities.
• To get GCD, multiply the common factors
• To get LCM, multiply highest factors.

## How to Find GCD and LCM of Two Polynomials - Questions

Question 1 :

Find the LCM and GCD for the following and verify that f (x) × g(x) = LCM × GCD

(i) 21x2y, 35xy2

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

Hence the relationship verified.

(ii)  (x3 −1)(x +1), (x3 +1)

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)

Hence the relationship verified.

(iii) (x2y + xy2), (x2 + xy)

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

Hence the relationship verified.

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