LCM OF ALGEBRAIC EXPRESSION

Least Common Multiple of Algebraic Expressions :

To find the least common multiple, first factorize the expressions if they can be factorized. Then find the product of the common factors and the rest of the factors.

If there are no common factors, then the least common multiple is the product of all the factors of the two expressions.

This product is the least common multiple of the given expressions. If the expressions are distinct and cannot be factorized, then the we should multiply everything.

Example  :

Find LCM of the following algebraic expressions.

(i)  2x2-18 y2, 5 x2y+15 xy2, x3+27y3

(ii)  (x+4)2 (x-3)3, (x-1) (x+4) (x-3)2

(iii)  10 (9x2+6xy+y2) , 12 (3x2-5xy-2y2), 14 (6x4+2x3)

(iv)  3(a-1), 2(a - 1)2 , (a2-1)

(i)   Answer :

2x2 - 18 y2, 5 x2y+15 xy2, x3+27y3

2x2 - 18 y =  2(x2- 9y2)

=  2(x2-(3y)2)

2x2 - 18 y2  =  2(x+3y) (x-3y) ----(1)

5x2y+15x  =  5xy(x+3y) ----(2)

x3+27y3  =  x3+(3y)3

=  (x+3y) (x2+x(3y)+(3y)2)

=  (x+3y) (x2+3xy+9y2)

= 2(x+3y) ⋅ ⋅  y  (x2+3xy+9y2)

=  10xy(x + 3y) (x2+3xy+9y2)

So, the required least common multiple is

10xy(x + 3y) (x2+3xy+9y2)

(ii)  Answer :

(x+4)2 (x-3)3, (x-1) (x+4) (x-3)2

By comparing (x+4) and (x+4)2, the highest term is (x+4)2.

By comparing (x-3)and (x-3)3, the highest term is (x-3)3

The extra term is (x-1).

So, the least common multiple is 

(x-1)(x+4)2(x-3)3

The least common multiple is 

(x-1)(x+4)2(x-3)3

(iii)  Answer :

10 (9x2+6xy+y2) , 12 (3x2-5xy-2y2), 14 (6x4+2x3)

10 (9x2+6xy+y2) :

10  =  2 ⋅ 5

By factoring 9x2+6xy+y2, we get

9x2+6xy+y =  9x2+3xy+3xy+y2

=  3x(3x+y)+y(3x+y)

(9x2+6xy+y2)  =  (3x+y)(3x+y)

10 (9x2+6xy+y2)  =   ⋅ 5 (9x2+6xy+y2) ----(1)

12(3x2-5xy-2y2) :

12  =  22 3

3x2-5xy-2y2  =  (3x2-6xy+xy-2y2)

=  3x(x-2y)+y(x-2y)

=  (3x+y) (x-2y) ----(2)

14(6x4+2x3) :

14  =  2  7

6x4+2x=  2x3(3x+1)

14(6x4+2x3)  =  2⋅ 7 x3 (3x+1) ----(3)

By comparing (1), (2) and (3), we get

=  22 ⋅  7  3  x³ ⋅ (3 x + y)²(3 x + 1)(x - 2y)

=  420 x3 (3 x + y)²(3 x + 1)(x - 2y)

So, the least common multiple is

420 x3 (3 x + y)2(3 x + 1)(x - 2y)

(iv)  Answer :

3(a-1), 2(a - 1)2 , (a2-1)

= 3 (a- 1) -------(1)

2 (a - 1)2  =  2(a-1)(a-1) -------(2)

(a2-1)  =  (a+1) (a-1) -------(3)

By comparing (1), (2) and (3), we get

=  3 ⋅ 2 (a - 1)2 (a + 1)

So, the least common multiple is 

6(a-1)2(a + 1)

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