Question

Find the least common multiple (LCM) of the expressions: x^2 + 2xy and x^3 - 4xy^2.

a) x^3 - 4xy^2
b) x^4 - 2x^2y
c) x^2(x + 2y)
d) x^3(x - 4y^2)

Answer 1

The LCM of x^2 + 2xy and x^3 - 4xy^2 is x^3(x - 2y)(x + 2y), which corresponds to option (d) x^3(x - 4y^2).

Step-by-step explanation:

To find the least common multiple (LCM) of the expressions x^2 + 2xy and x^3 - 4xy^2, we need to identify each term's factors and then determine the highest power of each factor present in either expression.

First, let's factor each expression:

For x^2 + 2xy, we can factor out an x:

• x(x + 2y)

For x^3 - 4xy^2, we can factor out an x as well:

• x(x^2 - 4y^2)

Notice that x^2 - 4y^2 is a difference of squares, which can be further factored into (x - 2y)(x + 2y). Hence, we rewrite the factorized form as:

• x(x - 2y)(x + 2y)

Now we determine the LCM. It must contain all factors present in either expression, raised to their highest powers. Here, the highest power of x is 3 (from x^3), and both x + 2y and (x - 2y) must be present.

The LCM is therefore:

• x^3 (x - 2y)(x + 2y)

Comparing the options given, x^3(x - 4y^2) matches our result, hence option (d) is the correct answer.