FINDING LCM OF ALGEBRAIC EXPRESSIONS WORKSHEET

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)  Solution :

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)  Solution :

(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)  Solution :

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)  Solution :

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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