LEAST COMMON MULTIPLE OF ALGEBRAIC EXPRESSIONS

Finding the LCM of polynomials

Problem 1 :

4xy², 2x²y³

Solution:

4xy² = 2² × x × y × y

2x²y³ = 2 × x × x × y × y × y

LCM = 2² ⋅ x² ⋅ y³

= 4x²y³

Problem 2 :

-9a³b, 12a²bc

Solution:

-9a³b = -3² × a × a × a × b

12a²bc = 2² × 3 × a × a × b × c

LCM = 3² ⋅ 2² ⋅ a³ ⋅ b ⋅ c

= 36a³bc

Problem 3 :

5xy, 15x²z, 10y²

Solution:

5xy = 5 × x × y

15x²z = 5 × 3 × x × x × z

10y² = 5 × 2 × y × y

LCM = 5 ⋅ 3 ⋅ 2 ⋅ x² ⋅ y² ⋅ z

= 30x²y²z

Problem 4 :

16m, -12m²n, 8n²

Solution:

16m = 2² × 2² × m

-12m²n = -2² × 3 × m × m × n

8n² = 2² × 2 × n × n

LCM = 2² × 2² × 3 × m² × n²

= 48m²n²

Problem 5 :

x, x - 2

Solution:

We don't see anything in common. So, the least common multiple will be the product of the given terms.

LCM = x(x - 2)

Problem 6 :

y², y + 3

Solution:

We don't see anything in common. So, the least common multiple will be the product of the given terms.

LCM = y²(y + 3)

Problem 7 :

x - 1, x + 4

Solution:

We don't see anything in common. So, the least common multiple will be the product of the given terms.

LCM = (x - 1) (x + 4)

Problem 8 :

z + 8, z + 2

Solution:

We don't see anything in common. So, the least common multiple will be the product of the given terms.

LCM = (z + 8) (z + 2)

Problem 9 :

x(x - 1), x², (x - 1)²

Solution:

By comparing x(x - 1) and (x - 1)², the highest term is (x - 1)².

The extra term is x².

So, the least common multiple is x²(x - 1)²

Problem 10 :

(y - 2)(y + 2), (y + 2)²

Solution:

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

The extra term is (y - 2).

So, the least common multiple is (y - 2)(y + 2)².

Problem 11 :

x² - 1, x² + 2x + 1

Solution:

x² - 1 = (x + 1) (x - 1)

x² + 2x + 1 = (x + 1)² 

So, the least common multiple is (x - 1)(x + 1)².

Problem 12 :

n² - 3n + 2, n² - 4

Solution:

n² - 3n + 2 = (n - 1)(n - 2)

n² - 4 = (n + 2)(n - 2)

So, the least common multiple is (n - 2)(n - 1)(n + 2).

Problem 13 :

t, t² - 1, t² + 5t - 6

Solution:

t² - 1 = (t + 1)(t - 1) 

t² + 5t - 6 = (t - 1)(t + 6)

So, the least common multiple is (t - 1)(t + 1)(t + 6).

Problem 14 :

w² - 9, 9w², w² - 6w + 9

Solution:

w² - 9 = (w + 3)(w - 3)

w² - 6w + 9 = (w - 3)²

The extra term is 9w².

So, the least common multiple is 9w² (w - 3)²(w + 3).

Problem 15 :

8x - 4, 6x² + x - 2

Solution:

8x - 4 = 4(2x - 1) ---> (1)

6x² + x - 2 = 6x² - 3x + 4x - 2

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

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

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

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

So, the least common multiple is 4(2x - 1)(3x + 2).

Problem 16 :

x³ - y³, x² - xy + y², x² - 2xy + y²

Solution:

x³ - y³ = (x - y)(x² + xy + y²)

x² - 2xy + y² = (x - y)²

The extra term is x² - xy + y².

So, the least common multiple is (x - y)²(x² - xy + y²)