To find the new volume of the cube after its side lengths are reduced, we start with the original side length of the cube, which is \(2x\). After the reduction, the new side length becomes \(2x - 3\).
We can use the Binomial Theorem to expand the volume of the cube:
\[ V = (2x - 3)^3 \]
According to the Binomial Theorem:
\[ (a - b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} (-b)^k \]
In our case, \(a = 2x\), \(b = 3\), and \(n = 3\). Thus, we apply the theorem:
\[ (2x - 3)^3 = \sum_{k=0}^{3} \binom{3}{k} (2x)^{3-k} (-3)^k \]
Let's calculate each term in the summation:
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For \(k = 0\): \[ \binom{3}{0} (2x)^3 (-3)^0 = 1 \cdot 8x^3 \cdot 1 = 8x^3 \]
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For \(k = 1\): \[ \binom{3}{1} (2x)^2 (-3)^1 = 3 \cdot 4x^2 \cdot (-3) = -36x^2 \]
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For \(k = 2\): \[ \binom{3}{2} (2x)^1 (-3)^2 = 3 \cdot 2x \cdot 9 = 54x \]
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For \(k = 3\): \[ \binom{3}{3} (2x)^0 (-3)^3 = 1 \cdot 1 \cdot (-27) = -27 \]
Now, we can put together all the terms:
\[ (2x - 3)^3 = 8x^3 - 36x^2 + 54x - 27 \]
Comparing this expression with the provided options, the correct expression for the new volume of the cube is:
\[ \boxed{8x^3 - 36x^2 + 54x - 27} \]