factor completely

1) 8x^3 + 27y^3

2) u^3 + 125v^3

Hope you have seen the response to the initial post.

To factor completely, we need to look for any common factors and then apply the appropriate factoring techniques. Let's factor the given expressions step by step:

1) To factor 8x^3 + 27y^3, we notice that both terms are perfect cubes. We can use the formula: a^3 + b^3 = (a + b)(a^2 - ab + b^2).

So, in this case, a = 2x and b = 3y. Let's apply the formula:

8x^3 + 27y^3 = (2x)^3 + (3y)^3
= (2x + 3y)((2x)^2 - (2x)(3y) + (3y)^2)
= (2x + 3y)(4x^2 - 6xy + 9y^2)

The expression 8x^3 + 27y^3 can be factored completely as (2x + 3y)(4x^2 - 6xy + 9y^2).

2) To factor u^3 + 125v^3, we can again use the formula: a^3 + b^3 = (a + b)(a^2 - ab + b^2).

Here, a = u and b = 5v. Applying the formula, we get:

u^3 + 125v^3 = (u)^3 + (5v)^3
= (u + 5v)((u)^2 - (u)(5v) + (5v)^2)
= (u + 5v)(u^2 - 5uv + 25v^2)

The expression u^3 + 125v^3 can be factored completely as (u + 5v)(u^2 - 5uv + 25v^2).

By using the appropriate formulas and factoring techniques, we have factorized the given expressions completely.