Question

Which of the following expressions is the factored form of x3+125y3
𝑥
3
+
125
𝑦
3
?(1 point)
Responses

(x+5y)(x2−5xy+25y2)
(
𝑥
+
5
𝑦
)
(
𝑥
2
−
5
𝑥
𝑦
+
25
𝑦
2
)
open paren x plus 5 y close paren times open paren x squared minus 5 x y plus 25 y squared close paren

(x−5y)(x2+5xy+25y2)
(
𝑥
−
5
𝑦
)
(
𝑥
2
+
5
𝑥
𝑦
+
25
𝑦
2
)
open paren x minus 5 y close paren times open paren x squared plus 5 x y plus 25 y squared close paren

(x2+5y2)(x2−5xy+25y2)
(
𝑥
2
+
5
𝑦
2
)
(
𝑥
2
−
5
𝑥
𝑦
+
25
𝑦
2
)
open paren x 2 plus 5 y squared close paren times open paren x squared minus 5 x y plus 25 y squared close paren

Answer 1

To factor the expression \( x^3 + 125y^3 \), we recognize it as a sum of cubes. The formula for factoring a sum of two cubes \( a^3 + b^3 \) is:

\[
a^3 + b^3 = (a + b)(a^2 - ab + b^2)
\]

In this case, we can identify:
- \( a = x \)
- \( b = 5y \) (since \( 125y^3 = (5y)^3 \))

Applying the sum of cubes formula:

1. \( a + b = x + 5y \)
2. \( a^2 = x^2 \)
3. \( ab = x(5y) = 5xy \)
4. \( b^2 = (5y)^2 = 25y^2 \)

Now we can assemble the factors:

\[
x^3 + 125y^3 = (x + 5y)(x^2 - 5xy + 25y^2)
\]

So the correct factored form of \( x^3 + 125y^3 \) is:

\[
(x + 5y)(x^2 - 5xy + 25y^2)
\]

Thus, the correct option is:

\[
\text{(x + 5y)(x² - 5xy + 25y²)}
\]