Answer :
To expand [tex]\((x-2)^6\)[/tex] using the binomial theorem, we need to understand that the binomial theorem provides a way to expand expressions of the form [tex]\((a + b)^n\)[/tex]. The theorem states that:
[tex]\[
(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k
\][/tex]
For [tex]\((x-2)^6\)[/tex], here [tex]\(a = x\)[/tex], [tex]\(b = -2\)[/tex], and [tex]\(n = 6\)[/tex]. Let's expand it step-by-step:
1. Identify the General Term:
The general term in the expansion is given by:
[tex]\[
\binom{6}{k} x^{6-k} (-2)^k
\][/tex]
2. Compute Individual Terms:
We calculate the terms from [tex]\(k = 0\)[/tex] to [tex]\(k = 6\)[/tex]:
- For [tex]\(k = 0\)[/tex]:
[tex]\[
\binom{6}{0} x^{6-0} (-2)^0 = 1 \cdot x^6 \cdot 1 = x^6
\][/tex]
- For [tex]\(k = 1\)[/tex]:
[tex]\[
\binom{6}{1} x^{6-1} (-2)^1 = 6 \cdot x^5 \cdot (-2) = -12x^5
\][/tex]
- For [tex]\(k = 2\)[/tex]:
[tex]\[
\binom{6}{2} x^{6-2} (-2)^2 = 15 \cdot x^4 \cdot 4 = 60x^4
\][/tex]
- For [tex]\(k = 3\)[/tex]:
[tex]\[
\binom{6}{3} x^{6-3} (-2)^3 = 20 \cdot x^3 \cdot (-8) = -160x^3
\][/tex]
- For [tex]\(k = 4\)[/tex]:
[tex]\[
\binom{6}{4} x^{6-4} (-2)^4 = 15 \cdot x^2 \cdot 16 = 240x^2
\][/tex]
- For [tex]\(k = 5\)[/tex]:
[tex]\[
\binom{6}{5} x^{6-5} (-2)^5 = 6 \cdot x^1 \cdot (-32) = -192x
\][/tex]
- For [tex]\(k = 6\)[/tex]:
[tex]\[
\binom{6}{6} x^{6-6} (-2)^6 = 1 \cdot 1 \cdot 64 = 64
\][/tex]
3. Combine All the Terms:
Collect all the terms together to form the expanded expression:
[tex]\[
x^6 - 12x^5 + 60x^4 - 160x^3 + 240x^2 - 192x + 64
\][/tex]
4. Match the Given Options:
Comparing this expansion with the given options, it matches option C:
[tex]\(x^6 - 12x^5 + 60x^4 - 160x^3 + 240x^2 - 192x + 64\)[/tex]
Therefore, the correct answer is option C.
[tex]\[
(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k
\][/tex]
For [tex]\((x-2)^6\)[/tex], here [tex]\(a = x\)[/tex], [tex]\(b = -2\)[/tex], and [tex]\(n = 6\)[/tex]. Let's expand it step-by-step:
1. Identify the General Term:
The general term in the expansion is given by:
[tex]\[
\binom{6}{k} x^{6-k} (-2)^k
\][/tex]
2. Compute Individual Terms:
We calculate the terms from [tex]\(k = 0\)[/tex] to [tex]\(k = 6\)[/tex]:
- For [tex]\(k = 0\)[/tex]:
[tex]\[
\binom{6}{0} x^{6-0} (-2)^0 = 1 \cdot x^6 \cdot 1 = x^6
\][/tex]
- For [tex]\(k = 1\)[/tex]:
[tex]\[
\binom{6}{1} x^{6-1} (-2)^1 = 6 \cdot x^5 \cdot (-2) = -12x^5
\][/tex]
- For [tex]\(k = 2\)[/tex]:
[tex]\[
\binom{6}{2} x^{6-2} (-2)^2 = 15 \cdot x^4 \cdot 4 = 60x^4
\][/tex]
- For [tex]\(k = 3\)[/tex]:
[tex]\[
\binom{6}{3} x^{6-3} (-2)^3 = 20 \cdot x^3 \cdot (-8) = -160x^3
\][/tex]
- For [tex]\(k = 4\)[/tex]:
[tex]\[
\binom{6}{4} x^{6-4} (-2)^4 = 15 \cdot x^2 \cdot 16 = 240x^2
\][/tex]
- For [tex]\(k = 5\)[/tex]:
[tex]\[
\binom{6}{5} x^{6-5} (-2)^5 = 6 \cdot x^1 \cdot (-32) = -192x
\][/tex]
- For [tex]\(k = 6\)[/tex]:
[tex]\[
\binom{6}{6} x^{6-6} (-2)^6 = 1 \cdot 1 \cdot 64 = 64
\][/tex]
3. Combine All the Terms:
Collect all the terms together to form the expanded expression:
[tex]\[
x^6 - 12x^5 + 60x^4 - 160x^3 + 240x^2 - 192x + 64
\][/tex]
4. Match the Given Options:
Comparing this expansion with the given options, it matches option C:
[tex]\(x^6 - 12x^5 + 60x^4 - 160x^3 + 240x^2 - 192x + 64\)[/tex]
Therefore, the correct answer is option C.
