Question

Yoshira applies the Binomial Theorem for the expansion of (2x−3)6 . Which of the following is the second term in the expansion? (1 point) Responses 2160x4 2160 x to the 4th power −576x5 negative 576 x to the 5th power −2160x4 negative 2160 x to the 4th power 576x5 576 x to the 5th power

Answer 1

To find the second term in the binomial expansion of \((2x - 3)^6\), we will use the Binomial Theorem. The theorem states that:

\[
(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k
\]

In this case, \(a = 2x\), \(b = -3\), and \(n = 6\).

The \(k\)-th term in the expansion is given by:

\[
T_k = \binom{n}{k} a^{n-k} b^k
\]

Here, we need the second term, which corresponds to \(k = 1\):

\[
T_2 = \binom{6}{1} (2x)^{6-1} (-3)^1
\]

Now we calculate each part:

1. **Binomial coefficient**:
\[
\binom{6}{1} = 6
\]

2. **Calculate \((2x)^{5}\)**:
\[
(2x)^{5} = 2^5 x^{5} = 32x^5
\]

3. **Calculate \((-3)^{1}\)**:
\[
(-3)^{1} = -3
\]

Now we substitute these values back into the term:

\[
T_2 = 6 \cdot 32x^5 \cdot (-3)
\]

Combine these values:

\[
T_2 = 6 \cdot 32 \cdot (-3) x^5 = 6 \cdot 32 \cdot (-3) x^5
\]
\[
= -576x^5
\]

Thus, the second term in the expansion of \((2x - 3)^6\) is \(\boxed{-576x^5}\).