Question

Yoshira applies the binomial theorem for the expansion of (2x-3)^6. Which of the following is the second term of the expansion? 576x^5, -2,160x^4, -576x^5, or 2,160x^4?

Answer 1

To find the second term of the expansion of \((2x - 3)^6\) using the binomial theorem, we can utilize the formula for the \(k\)-th term in the expansion:

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

where:
- \(n\) is the exponent (in this case, \(n=6\)),
- \(a\) is the first term in the binomial expression (here, \(a=2x\)),
- \(b\) is the second term in the binomial expression (here, \(b=-3\)), and
- \(k\) is the term number we are calculating (recall that \(k\) starts at 0).

The second term corresponds to \(k=1\). Thus, we will calculate \(T_1\):

\[
T_1 = \binom{6}{1} (2x)^{6-1} (-3)^1
\]

Calculating each component:

1. The binomial coefficient:
\[
\binom{6}{1} = 6
\]

2. The term \((2x)^{5}\):
\[
(2x)^5 = 32x^5
\]

3. The term \((-3)^1\):
\[
(-3) = -3
\]

Now, substitute these values back into the term formula:

\[
T_1 = 6 \cdot 32x^5 \cdot (-3)
\]

Now perform the multiplication:

\[
T_1 = 6 \cdot 32 \cdot (-3) x^5
\]
\[
T_1 = 192 \cdot (-3) x^5
\]
\[
T_1 = -576 x^5
\]

Thus, the second term of the expansion is:

\[
\boxed{-576x^5}
\]