To solve this problem, we need to expand and verify the polynomial expression [tex]\(125 x^3 - 75 x^2 y + 15 x y^2 - y^3\)[/tex] to determine if it matches the form [tex]\((a - b)^3\)[/tex], where [tex]\(a = 5x\)[/tex] and [tex]\(b = y\)[/tex].
Let's start by using the binomial expansion formula for [tex]\((a - b)^3\)[/tex], which is:
[tex]\[
(a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3
\][/tex]
Here, we substitute [tex]\(a = 5x\)[/tex] and [tex]\(b = y\)[/tex].
1. Calculate [tex]\(a^3\)[/tex]:
[tex]\[
(5x)^3 = 125x^3
\][/tex]
2. Calculate [tex]\(-3a^2b\)[/tex]:
[tex]\[
-3(5x)^2y = -3 \cdot 25x^2y = -75x^2y
\][/tex]
3. Calculate [tex]\(3ab^2\)[/tex]:
[tex]\[
3(5x)y^2 = 3 \cdot 5x \cdot y^2 = 15xy^2
\][/tex]
4. Calculate [tex]\(-b^3\)[/tex]:
[tex]\[
-y^3 = -y^3
\][/tex]
Now, add these components together:
[tex]\[
(5x - y)^3 = 125x^3 - 75x^2 y + 15x y^2 - y^3
\][/tex]
Therefore, the expanded expression:
[tex]\[
125x^3 - 75x^2 y + 15x y^2 - y^3
\][/tex]
is indeed equivalent to [tex]\((5x - y)^3\)[/tex].
