To expand the expression \((2x - 1)^3\), we will use the binomial expansion formula, which states that
\[ (a - b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} (-b)^k \]
In this case, \(a = 2x\), \(b = 1\), and \(n = 3\).
Using the binomial theorem:
\[ (2x - 1)^3 = \sum_{k=0}^{3} \binom{3}{k} (2x)^{3-k} (-1)^k \]
We will calculate each term separately for \(k = 0, 1, 2, 3\):
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For \(k = 0\): \[ \binom{3}{0} (2x)^3 (-1)^0 = 1 \cdot (2x)^3 \cdot 1 = 8x^3 \]
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For \(k = 1\): \[ \binom{3}{1} (2x)^2 (-1)^1 = 3 \cdot (2x)^2 \cdot (-1) = 3 \cdot 4x^2 \cdot (-1) = -12x^2 \]
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For \(k = 2\): \[ \binom{3}{2} (2x)^1 (-1)^2 = 3 \cdot (2x) \cdot 1 = 6x \]
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For \(k = 3\): \[ \binom{3}{3} (2x)^0 (-1)^3 = 1 \cdot 1 \cdot (-1) = -1 \]
Now we combine all the terms together: \[ (2x - 1)^3 = 8x^3 - 12x^2 + 6x - 1 \]
Thus, the expanded form of \((2x - 1)^3\) in standard form is:
\[ \boxed{8x^3 - 12x^2 + 6x - 1} \]
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