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)
8x^3−36x^2+54x+27
8x^3+36x^2+54x−27
8x^3−36x^2+54x−27
8x^3+36x^2+54x+27
The correct expression for the new volume of the cube can be determined by expanding the expression for the volume of the original cube, and then substituting (2x - 3) for each side length.
The original volume of the cube is given by (2x)^3 = 8x^3.
Substituting (2x - 3) for each side length, we get:
(2x - 3)(2x - 3)(2x - 3) = (2x - 3)^3
Using the Binomial Theorem, this expression expands to:
8x^3 - 36x^2 + 54x - 27.
Therefore, the correct expression for the new volume of the cube is:
8x^3 - 36x^2 + 54x - 27.
So, the correct answer is: 8x^3 - 36x^2 + 54x - 27.
To find the new volume of the cube after it has been shrunk, we need to use the binomial theorem to expand the expression (2x - 3)^3.
The binomial theorem states that for any binomial expression (a + b)^n, the expanded form can be found using the formula:
(a + b)^n = C(n, 0)a^n b^0 + C(n, 1)a^(n-1) b^1 + C(n, 2)a^(n-2) b^2 + ... + C(n, n-1)a^1 b^(n-1) + C(n, n)a^0 b^n
In this case, a = 2x and b = -3, and we need to expand (2x - 3)^3.
Plugging these values into the formula, we get:
(2x - 3)^3 = C(3, 0)(2x)^3 (-3)^0 + C(3, 1)(2x)^2 (-3)^1 + C(3, 2)(2x)^1 (-3)^2 + C(3, 3)(2x)^0 (-3)^3
Now, let's expand each term:
C(3, 0)(2x)^3 (-3)^0 = 1 * (2x)^3 * 1 = 8x^3
C(3, 1)(2x)^2 (-3)^1 = 3 * (2x)^2 * (-3) = -36x^2
C(3, 2)(2x)^1 (-3)^2 = 3 * (2x)^1 * 9 = 54x
C(3, 3)(2x)^0 (-3)^3 = 1 * 1 * (-27) = -27
Now, combining all the terms, we get:
8x^3 - 36x^2 + 54x - 27
Therefore, the correct expression for the new volume of the cube is:
8x^3 - 36x^2 + 54x - 27.
So, the correct answer is option C:
8x^3 - 36x^2 + 54x - 27.