Question

Expand the expression to a polynomial in standard form:
left parenthesis, 2, x, minus, 1, right parenthesis, cubed
(2x−1)
3

Answer 1

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 \(k = 0, 1, 2, 3\):

1. **For \(k = 0\)**:
\[
\binom{3}{0} (2x)^3 (-1)^0 = 1 \cdot (2x)^3 \cdot 1 = 8x^3
\]

2. **For \(k = 1\)**:
\[
\binom{3}{1} (2x)^2 (-1)^1 = 3 \cdot (2x)^2 \cdot (-1) = 3 \cdot 4x^2 \cdot (-1) = -12x^2
\]

3. **For \(k = 2\)**:
\[
\binom{3}{2} (2x)^1 (-1)^2 = 3 \cdot (2x) \cdot 1 = 6x
\]

4. **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:

\[
\boxed{8x^3 - 12x^2 + 6x - 1}
\]