LCM OF ALGEBRAIC EXPRESSIONS

Find LCM of the following algebraic expressions.

Example 1 :

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

Solution :

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)

Example 2 :

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

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)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

Example 3 :

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

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 + 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)

Example 4 :

3(a-1), 2(a - 1)² , (a²-1)

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)

Example 5 :

8x⁴ y², 48x⁴ y⁴

Solution :

8x⁴ y² = 2 ⋅ 2 ⋅ 2 ⋅ x⁴ y²

= 2³ ⋅ x⁴ y²-------(1)

48x⁴ y⁴ = 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ x⁴ y²

=  2⁴ ⋅ x⁴ y⁴-------(2)

Comparing (1) and (2), 

least common multiple =  2⁴ ⋅ x⁴ y⁴

= 16x⁴y⁴

Example 6 :

5x - 10, 5x²- 20

Solution :

= 5x - 10

Factoring 5, we get

= 5(x - 2) --------(1)

5x²- 20

Factoring 5, we get

= 5(x - 4) --------(2)

Comparing (1) and (2), we get

least common multiple = 5(x - 2) (x - 4)

Example 7 :

x⁴ - 1, x² - 2x + 1

Solution :

= x⁴ - 1

= (x²)² - 1

Looks like an algebraic identity, 

= (x² + 1) (x² - 1²)

= (x² + 1) (x + 1)(x - 1)  ---(1)

x² - 2x + 1 = (x - 1) (x - 1)  ---(2)

Least common multiple = (x - 1) (x - 1)(x² + 1)

= (x⁴ - 1)

Example 8 :

x³ - 27, (x - 3)² , x² - 9

Solution :

x³ - 27

Looks like an algebraic identity, a³ - b³

a³ - b³ = (a - b)(a² - ab + b²)

x³ - 3³ = (x - 3)(x² - x(3) + 3²)

= (x - 3)(x² - 3x + 9) -----(1)

(x - 3)² = (x - 3)(x - 3) -----(2)

x² - 9 =  x² - 3²

= (x + 3)(x - 3) -----(3)

Comparing (1), (2) and (3), we get

(x - 3)(x + 3) (x² - 3x + 9)

Example 9 :

x² - 3p + 2, p² - 4

Solution :

x² - 3p + 2

Using factoring, we find the linear factors

= (p - 1) (p - 2) ---(1)

p² - 4 = p² - 2²

=(p + 2)(p - 2) ----(2)

Comparing (1) and (2), we get

least common multiple = (p + 2 )(p - 1) (p - 2)

Example 10 :

2x² - 5x - 3, 4x² - 36

Solution :

2x² - 5x - 3 = 2x² - 6x + 1x - 3

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

= (2x + 1)(x - 3) ---(1)

4x² - 36 = 4(x² - 9)

= 4(x² - 3²)

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

Comparing (1) and (2), we get

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

So, the least common multiple is 

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

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