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------------------------------------------------ A cube is shrunk so that its side lengths of [tex]2x[/tex] are reduced by 3 units. Using the Binomial Theorem, which of the following is the correct expression for the new volume of the cube?

A. [tex]8x^3 + 36x^2 + 54x + 27[/tex]

B. [tex]8x^3 + 36x^2 + 54x - 27[/tex]

C. [tex]8x^3 - 36x^2 + 54x - 27[/tex]

D. [tex]8x^3 - 36x^2 + 54x + 27[/tex]

To find the new volume of the cube after reducing its side length, let's follow these steps:

1. Understand the Problem:
- Originally, the cube has side lengths of [tex]\(2x\)[/tex].
- Each side length is reduced by 3 units. So, the new side length of the cube becomes [tex]\((2x - 3)\)[/tex].

2. Calculate the New Volume:
- The formula for the volume of a cube is [tex]\((\text{side})^3\)[/tex].
- Thus, the new volume will be [tex]\((2x - 3)^3\)[/tex].

3. Expand the Expression Using the Binomial Theorem:
- [tex]\((2x - 3)^3\)[/tex] can be expanded using the Binomial Theorem which states that [tex]\((a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3\)[/tex].

Applying this to [tex]\((2x - 3)^3\)[/tex]:
- Let [tex]\(a = 2x\)[/tex] and [tex]\(b = 3\)[/tex].

[tex]\[
\begin{align*}
(2x - 3)^3 & = (2x)^3 - 3(2x)^2 \cdot 3 + 3(2x) \cdot 3^2 - 3^3 \\
& = 8x^3 - 3 \cdot 4x^2 \cdot 3 + 3 \cdot 2x \cdot 9 - 27 \\
& = 8x^3 - 36x^2 + 54x - 27
\end{align*}
\][/tex]

4. Compare with the Given Options:
- The expanded expression is [tex]\(8x^3 - 36x^2 + 54x - 27\)[/tex].

Therefore, the correct expression for the new volume of the cube is:
[tex]\[ 8x^3 - 36x^2 + 54x - 27 \][/tex]

So, the correct option is: [tex]\(8x^3 - 36x^2 + 54x - 27\)[/tex].