Answer :
D) [tex]8x^3 - 36x^2 + 54x - 27[/tex] is the correct expression for the new volume of the cube.
To solve this problem, we need to find the new volume of a cube when its side lengths are reduced by 3 units from [tex]2x[/tex], using the Binomial Theorem.
Original Side Length: The original side length of the cube is [tex]2x[/tex].
New Side Length: After the reduction by 3 units, the new side length becomes [tex]2x - 3[/tex].
Volume Formula: The volume [tex]V[/tex] of a cube is given by [tex]V = \text{side length}^3[/tex]. So the new volume is:
[tex](2x - 3)^3[/tex]
Binomial Expansion: According to the Binomial Theorem, [tex](a - b)^3 = a^3 - 3a^2b + 3ab^2 - b^3[/tex]. Here [tex]a = 2x[/tex] and [tex]b = 3[/tex]:
[tex]a^3 = (2x)^3 = 8x^3[/tex]
[tex]-3a^2b = -3(2x)^2(3) = -36x^2[/tex]
[tex]3ab^2 = 3(2x)(3^2) = 54x[/tex]
[tex]-b^3 = -(3)^3 = -27[/tex]
Combine the Terms: Therefore, the expanded form is:
[tex]8x^3 - 36x^2 + 54x - 27[/tex]
