Answer :
To find the second term in the expansion of [tex]\((2x + 3)^4\)[/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], where [tex]\(\binom{n}{k}\)[/tex] is a binomial coefficient.
In this case, [tex]\(a = 2x\)[/tex], [tex]\(b = 3\)[/tex], and [tex]\(n = 4\)[/tex]. We want to determine the second term of the expansion, which corresponds to [tex]\(k = 1\)[/tex].
Here are the steps to find the second term:
1. Identify the Binomial Coefficient:
The binomial coefficient for the second term (when [tex]\(k = 1\)[/tex]) is [tex]\(\binom{4}{1}\)[/tex]. This is calculated as:
[tex]\(\binom{4}{1} = 4\)[/tex].
2. Determine the Powers:
- The power of [tex]\(a = 2x\)[/tex] will be [tex]\(n-k = 4-1 = 3\)[/tex].
- The power of [tex]\(b = 3\)[/tex] will be [tex]\(k = 1\)[/tex].
3. Calculate the Second Term:
Multiply the binomial coefficient by the powers of [tex]\(a\)[/tex] and [tex]\(b\)[/tex]:
[tex]\(\binom{4}{1} \cdot (2x)^{3} \cdot 3^{1}\)[/tex]
[tex]\(= 4 \cdot (2x)^{3} \cdot 3\)[/tex]
4. Simplify the Expression:
- [tex]\((2x)^{3} = 8x^{3}\)[/tex]
- So, multiply everything:
[tex]\(4 \cdot 8x^{3} \cdot 3 = 96x^{3}\)[/tex]
Thus, the second term in the expansion of [tex]\((2x + 3)^4\)[/tex] is [tex]\(96x^{3}\)[/tex].
Given the choices:
a. [tex]\(36 x^{3}\)[/tex]
b. [tex]\(24 x^{3}\)[/tex]
c. [tex]\(12 x^{3}\)[/tex]
d. [tex]\(48 x^{3}\)[/tex]
None of these match the correctly calculated second term [tex]\(96x^{3}\)[/tex]. Therefore, there seems to be a mistake in the provided options, as [tex]\(96x^{3}\)[/tex] is the correct second term.
In this case, [tex]\(a = 2x\)[/tex], [tex]\(b = 3\)[/tex], and [tex]\(n = 4\)[/tex]. We want to determine the second term of the expansion, which corresponds to [tex]\(k = 1\)[/tex].
Here are the steps to find the second term:
1. Identify the Binomial Coefficient:
The binomial coefficient for the second term (when [tex]\(k = 1\)[/tex]) is [tex]\(\binom{4}{1}\)[/tex]. This is calculated as:
[tex]\(\binom{4}{1} = 4\)[/tex].
2. Determine the Powers:
- The power of [tex]\(a = 2x\)[/tex] will be [tex]\(n-k = 4-1 = 3\)[/tex].
- The power of [tex]\(b = 3\)[/tex] will be [tex]\(k = 1\)[/tex].
3. Calculate the Second Term:
Multiply the binomial coefficient by the powers of [tex]\(a\)[/tex] and [tex]\(b\)[/tex]:
[tex]\(\binom{4}{1} \cdot (2x)^{3} \cdot 3^{1}\)[/tex]
[tex]\(= 4 \cdot (2x)^{3} \cdot 3\)[/tex]
4. Simplify the Expression:
- [tex]\((2x)^{3} = 8x^{3}\)[/tex]
- So, multiply everything:
[tex]\(4 \cdot 8x^{3} \cdot 3 = 96x^{3}\)[/tex]
Thus, the second term in the expansion of [tex]\((2x + 3)^4\)[/tex] is [tex]\(96x^{3}\)[/tex].
Given the choices:
a. [tex]\(36 x^{3}\)[/tex]
b. [tex]\(24 x^{3}\)[/tex]
c. [tex]\(12 x^{3}\)[/tex]
d. [tex]\(48 x^{3}\)[/tex]
None of these match the correctly calculated second term [tex]\(96x^{3}\)[/tex]. Therefore, there seems to be a mistake in the provided options, as [tex]\(96x^{3}\)[/tex] is the correct second term.
