To determine the coefficient of the [tex]\( x^5 \)[/tex] term in the expansion of [tex]\( (2x + 3)^7 \)[/tex], we can use the binomial theorem. The binomial theorem states that:
[tex]\[
(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k
\][/tex]
In this case, the expression is [tex]\( (2x + 3)^7 \)[/tex]. Here, [tex]\( a = 3 \)[/tex], [tex]\( b = 2x \)[/tex], and [tex]\( n = 7 \)[/tex].
We are interested in the coefficient of the [tex]\( x^5 \)[/tex] term. This term corresponds to [tex]\( b^5 \)[/tex], so we will use [tex]\( b = 2x \)[/tex] in our calculations.
1. Identify the appropriate term:
[tex]\[
T(k) = \binom{n}{k} a^{n-k} b^k
\][/tex]
For the [tex]\( x^5 \)[/tex] term, [tex]\( k = 5 \)[/tex].
2. Calculate each component:
- Binomial coefficient [tex]\( \binom{7}{5} \)[/tex]:
[tex]\[
\binom{7}{5} = 21
\][/tex]
- Calculate [tex]\( a^{n-k} \)[/tex], where [tex]\( a = 3 \)[/tex] and [tex]\( n - k = 2 \)[/tex]:
[tex]\[
3^{2} = 9
\][/tex]
- Calculate [tex]\( b^k \)[/tex], where [tex]\( b = 2x \)[/tex] and [tex]\( k = 5 \)[/tex]:
[tex]\[
(2x)^5 = 2^5 \cdot x^5 = 32 x^5
\][/tex]
3. Combine these elements to form the term [tex]\( T(5) \)[/tex]:
[tex]\[
T(5) = \binom{7}{5} \cdot 3^2 \cdot 32 x^5
\][/tex]
4. Evaluate the coefficient:
[tex]\[
T(5) = 21 \cdot 9 \cdot 32 x^5
\][/tex]
[tex]\[
T(5) = 6048 x^5
\][/tex]
Thus, the coefficient of the [tex]\( x^5 \)[/tex] term in the expansion of [tex]\( (2x + 3)^7 \)[/tex] is:
[tex]\[
\boxed{6048}
\][/tex]
