# LCM OF POLYNOMIALS BY FACTORING

lcm of Polynomials by Factoring :

The Least Common Multiple of two or more algebraic expressions is the expression of lowest degree (or power) such that the expressions exactly divide it.

To find LCM by factorization method

(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.

## LCM of Polynomials by Factoring Examples

Question 1 :

Find the LCM of the given expressions.

(i) 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

(ii) -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

(iii)  16m, -12m2n2, 8n2

Solution :

16m  =  2⋅ m

-12m2n =  -2⋅ 3 m2n2

8n2 =  2⋅ n

L.C.M  =  2m2n2

=  16m2n2

(iv) 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)

(v)  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)

(vi) (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) After having gone through the stuff given above, we hope that the students would have understood, "lcm of Polynomials by Factoring".

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