LCM OF ALGEBRAIC EXPRESSION

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 y²  =  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) 5 ⋅ x ⋅  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)² 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+y²  =  9x²+3xy+3xy+y²

=  3x(3x+y)+y(3x+y)

(9x²+6xy+y²)  =  (3x+y)(3x+y)

10 (9x²+6xy+y²)  =   2 ⋅ 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+2x³ =  2x³(3x+1)

14(6x⁴+2x³)  =  2² ⋅ 7⋅ x³ (3x+1) ----(3)

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

=  2² ⋅ 5 ⋅ 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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