High School

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]

Answer :

To solve the problem of finding the new volume of a cube after its side lengths have been reduced, we can use the Binomial Theorem to expand the expression for the new side length.

1. Original Side Length: The original side length of the cube is [tex]\(2x\)[/tex].

2. Reduction: We are told that each side length is reduced by 3 units. Therefore, the new side length of the cube is [tex]\((2x - 3)\)[/tex].

3. Expression for New Volume: The volume of a cube is given by the cube of its side length. So, the volume of the cube with the reduced side length is [tex]\((2x - 3)^3\)[/tex].

4. Use the Binomial Theorem to Expand [tex]\((2x - 3)^3\)[/tex]:

The Binomial Theorem states:
[tex]\[
(a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3
\][/tex]

Applying this formula to our expression:

- Let [tex]\(a = 2x\)[/tex] and [tex]\(b = 3\)[/tex].
- [tex]\(a^3 = (2x)^3 = 8x^3\)[/tex]
- [tex]\(-3a^2b = -3(2x)^2(3) = -3 \cdot 4x^2 \cdot 3 = -36x^2\)[/tex]
- [tex]\(3ab^2 = 3(2x)(3)^2 = 3 \cdot 2x \cdot 9 = 54x\)[/tex]
- [tex]\(-b^3 = -(3)^3 = -27\)[/tex]

5. Combine Terms:

Combine all the terms obtained from the expansion:
[tex]\[
(2x - 3)^3 = 8x^3 - 36x^2 + 54x - 27
\][/tex]

The expanded expression for the new volume of the cube is:

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

Thus, the correct expression for the new volume of the cube is [tex]\(\boxed{8x^3 - 36x^2 + 54x - 27}\)[/tex].