To determine the term in the expansion of [tex]\((x + 2)^6\)[/tex] that is independent of [tex]\(x\)[/tex], we need to use the binomial theorem, which states:
[tex]\[
(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k
\][/tex]
Here, [tex]\(a = x\)[/tex], [tex]\(b = 2\)[/tex], and [tex]\(n = 6\)[/tex]. The general term in the expansion can be written as:
[tex]\[
\binom{6}{k} x^{6-k} \cdot 2^k
\][/tex]
We seek the term that is independent of [tex]\(x\)[/tex], which means we need [tex]\(x\)[/tex] to have an exponent of 0. This occurs when the exponent of [tex]\(x\)[/tex] in the general term [tex]\(x^{6-k}\)[/tex] is zero:
[tex]\[
6 - k = 0
\][/tex]
Solving for [tex]\(k\)[/tex]:
[tex]\[
k = 6
\][/tex]
At [tex]\(k = 6\)[/tex], the term becomes:
[tex]\[
\binom{6}{6} x^{6-6} \cdot 2^6 = \binom{6}{6} x^0 \cdot 2^6 = 1 \cdot 1 \cdot 64 = 64
\][/tex]
This confirms that the term independent of [tex]\(x\)[/tex] is simply 64, which matches our result.
The term in a binomial expansion [tex]\(\sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k \)[/tex] is indexed from [tex]\(k=0\)[/tex] up to [tex]\(k=n\)[/tex]. Since our [tex]\(k=6\)[/tex] corresponds to the term with [tex]\(k+1 = 7\)[/tex].
Therefore, the term independent of [tex]\(x\)[/tex] is the 7th term. However, since the numbering of the options given are incorrect based on the structured Python conclusion, we can state the follow up.
__None of the choices (A,B,C,D) correctly identify the term related to independent x.__
If instead the Python output explicitly stated an options layout which involves 7 we would consider:
Hence, based on our detailed manual binomial step conclusion unrelated to Python,
The correct answer for identification as independent of [tex]\(x\)[/tex] would be:
`7th term` as accurate placement.
Thus for this layout:
The term independent of [tex]\(x\)[/tex] in the expansion [tex]\((x + 2)^6\)[/tex] is not listed in the provided choices (3rd, 4th, 5th, or 6th term).
