# HOW TO FIND LCM FROM TWO POLYNOMIALS AND GCD

How to Find lcm From Two Polynomials and gcd ?

The product of two polynomials is the product of their LCM and GCD. That is,

f (x) × g(x) = LCM × GCD

## How to Find lcm From Two Polynomials and gcd- Questions

Question 1 :

Find the LCM of each pair of the following polynomials

(i) a2 + 4a −12, a2 −5a + 6 whose GCD is a -2

Solution :

Let f(x) = a2 + 4a −12, g(x) = a2 −5a + 6

 f(x)  =   a2 + 4a −12    =  a2 + 6a - 2a −12  =  a(a + 6) - 2(a + 6) f(x)  =  (a + 6)(a - 2) g(x) = a2 −5a + 6  =  a2 - 2a - 3a + 6  =  a(a - 2) - 3(a - 6)g(x)  =  (a - 3)(a - 2)

GCD is a -2

f (x) × g(x) = LCM × GCD

LCM  =  [f(x) × g(x)] / GCD

LCM  =  [(a + 6)(a - 2)  (a - 3)(a - 2)] /  a -2

LCM  =  (a + 6)(a - 2)  (a - 3)

(ii) x 4 -27a3x, (x -3a)2 whose GCD is (x -3a)

Solution :

Let f(x) = x 4 -27a3x, g(x)  =   (x -3a)2

 f(x)  =  x(x3 - 27a3)=  x(x3-(3a)3)  = x (x-3a)(x2-x(3a)+(3a)2)f(x) = x (x- 3a)(x2-3ax+9a2) g(x)  =   (x -3a)2

GCD is (x -3a)

f (x) × g(x) = LCM × GCD

LCM  =  [f(x) × g(x)] / GCD

LCM  =  [x (x- 3a)(x2-3ax+9a2) (x -3a)2] /  (x -3a)

LCM  =  x(x2-3ax+9a2) (x -3a)2

## How to Find GCD from Two Polynomials and LCM

Question 1 :

Find the GCD of each pair of the following polynomials

(i) 12(x4 -x3), 8(x4 −3x3 +2x2) whose LCM is 24x3(x -1)(x -2)

Solution :

Let f(x)  =  12(x4 -x3), g(x)  =  8(x4 −3x3 +2x2)

LCM  =  24x3(x -1)(x-2)

 f(x)  =  12(x4 -x3)f(x)  =  12x3(x - 1) g(x)  =  8(x4 −3x3 +2x2)g(x)  =  8x2(x2 - 3x + 2)

GCD  =  24x3(x -1)(x -2)

f (x) × g(x) = LCM × GCD

GCD  =  [f(x) × g(x)] / LCM

GCD  =  [12x3(x - 1) 8x2(x2 - 3x + 2)]/ 24x3(x -1)(x -2)

GCD  =  4x2(x-1)

(ii) (x3 + y3), (x4 + x2y2 + y4) whose LCM is (x3 + y3)(x2 + xy + y2)

Solution :

Let f(x)  =  (x3 + y3), g(x)  =  (x4 + x2y2 + y4)

LCM is (x3 + y3)(x2 + xy + y2)

 f(x)  =  (x3 + y3) g(x)  =  (x4 + x2y2 + y4)= (x2 + y2)2 - (xy)2= (x2 + y2)2 - (xy)2= (x2-xy+ y2 )(x2+ xy+ y2)

LCM  =  (x3 + y3)(x2 + xy + y2)

f (x) × g(x) = LCM × GCD

GCD  =  [f(x) × g(x)] / LCM

GCD  =  [ (x3 + y3)(x2-xy+ y)(x2+ xy+ y2)] / (x3 + y3)(x2 + xy + y2)

GCD  =  (x- xy +  y)

Question 3 :

Given the LCM and GCD of the two polynomials p(x) and q(x) find the unknown polynomial in the following table

 (i)LCM = a3 −10a2 +11a + 70GCD  =  a - 7p(x)  =  a2 −12a + 35 find q(x) (ii)LCM = (x2 +y2)(x4 +x2y2+y4)GCD  =  (x2 -y2) (x4 −y4)q(x)  =  (x2 +y2 −xy)find p(x)

Solution :

(i)

p (x) × q(x) = LCM × GCD

q(x)  =  (LCM × GCD)/p (x)

=   (a3 −10a2 +11a + 70)( a - 7)/(a2 −12a + 35) =  (a + 2) (a - 7)

(ii)  LCM = (x2 +y2)  (x4 +x2y2+y4)

GCD  =  (x2 -y2

q(x)  = (x4 −y4) (x2 +y2 −xy)

find p(x)

p(x)  =  (LCM × GCD)/q(x)

=  (x2 +y2)  (x4 +x2y2+y4)(x2 -y2) /(x4 −y4)(x2 +y2 −xy)  ---(1)

(x4 −y4)  =  (x2 + y2)(x2 - y2)

(x4 + x2y2 + y4= (x2 + y2)2 - (xy)2

= (x2 + y2)2 - (xy)2

= (x2-xy+ y)(x2+ xy+ y2)

By applying these values in (1), we get

=   (x2-xy+ y2)(x2+ xy+ y2)/(x2 +y2 −xy)

=  (x2+ xy+ y2) After having gone through the stuff given above, we hope that the students would have understood, "How to Find lcm From Two Polynomials and gcd".

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