# LCM OF POLYNOMIALS BY FACTORING

The least common multiple of two or more polynomials is the expression of lowest degree (or power) such that the polynomials are exactly divided by it.

The following steps will be useful to find least common multiple of two or more polynomials by factoring.

(i) Each expression is first resolved into its factors.

(ii) The highest power of the factors will be the LCM.

(iii) If the expressions have numerical coefficients, find their LCM.

(iv) The product of the LCM of factors and coefficient is the required LCM.

Find the least common multiple of the following polynomials.

Question 1 :

4x2y, 8x3y2

Solution :

Let us factor each of the polynomials.

 4x2y  =  2 ⋅ 2 ⋅ x2 ⋅ y  =  22 ⋅ x2 ⋅ y 8x3y2  =  2 ⋅ 2 ⋅ 2⋅ x3 ⋅ y2=  23 ⋅ x3 ⋅ y2

The highest power for 2 is 22, x is x3 and y is y2

L.C.M  =  23⋅ x⋅ y2   =  8 xy2

Hence the answer is xy2.

Question 2 :

-9a3b2, 12a2b2c

Solution :

Let us factor each of the polynomials

 -9a3b2    =  -3 ⋅ 3 ⋅ a3 ⋅ b2  =  22 ⋅ a3 ⋅ b2 12a2b2c  =  2 ⋅ 2 ⋅ 3⋅ a2 ⋅ b2⋅ c=  22 ⋅ 3 ⋅ a2 ⋅ b2⋅ c

L.C.M  =  22 ⋅ 3  a⋅ b2⋅ c

=  12a2 b2c

Question 3 :

16m, -12m2n2, 8n2

Solution :

16m  =  2⋅ m

-12m2n =  -2⋅ 3 m2n2

8n2 =  2⋅ n

L.C.M  =  2m2n2

=  16m2n2

Question 4 :

p2 − 3p +2, p2 - 4

Solution :

p2 − 3p + 2  =  p2 − p - 2p + 2

=  p(p - 1) - 2(p - 1)

=  (p - 1)(p - 2)

p2 - 4  =  p2 - 22

=  (p + 2) (p - 2)

L.C.M  =  (p+2)(p-2)(p-1)

Question 5 :

2x2 - 5x -3, 4x2 -36

Solution :

2x2 - 5x -3  =  2x2 - 6x + 1x -3

=  2x (x - 3) + 1(x - 3)

=  (2x + 1)(x - 3)

4x2 - 36  =  4(x2 - 9)

=  4(x2 - 3)2

=  4(x - 3)(x + 3)

L.C.M  =  4(2x + 1)(x + 3)(x - 3)

Question 6 :

(2x2 -3xy)2, (4x -6y)3, 8x3 -27y3

Solution :

(2x2 -3xy) =  [x(2x - 3y)]2

=  x2(2x - 3y)2

(4x -6y)=  [2(2x -3y)]3

=  8(2x -3y)3

8x3 -27y =  23x3 - 33 y3

=  (2x)3 - (3y)3

=  (2x - 3y) ((2x)2 - (2x)(3y) + (3y)2)

=  (2x - 3y) (4x2 - 6xy + 9y2)

L.C.M  =  8x2(2x -3y)3(4x2 - 6xy + 9y2)

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