Question

A cube is shrunk so that its side lengths of 2x are reduced by 3 units. Using the Binomial Theorem, which of the following is the correct expression for the new volume of the cube?(1 point)

a. 8x^3+36x^2+54x+27
b. 8x^3-36x^2+54x+27
c. 8x^3+36x^2+54x-27
d. 8x^3-36x^2+54x-27

Answer 1

To find the new volume of the cube, we need to calculate the cube of the new side length. Let's use the binomial theorem for that.

The new side length is 2x reduced by 3 units, which gives us (2x - 3).

Using the binomial theorem, we can expand (2x - 3)^3 as:

(2x - 3)^3 = (2x)^3 - 3(2x)^2(3) + 3(2x)(3^2) - 3^3

Simplifying this expression:

8x^3 - 24x^2 + 36x - 27

Therefore, the correct expression for the new volume of the cube is:

8x^3 - 24x^2 + 36x - 27

Option a. 8x^3 + 36x^2 + 54x + 27 is incorrect.
Option b. 8x^3 - 36x^2 + 54x + 27 is incorrect.
Option c. 8x^3 + 36x^2 + 54x - 27 is incorrect.
Option d. 8x^3 - 36x^2 + 54x - 27 is correct.