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Answer :
To expand [tex]\((x - 2)^6\)[/tex] using the Binomial Theorem, we follow these steps:
1. Understand 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.
2. Identify variables: In this expression, [tex]\(a = x\)[/tex], [tex]\(b = -2\)[/tex], and [tex]\(n = 6\)[/tex].
3. Apply the Binomial Theorem: We need to expand [tex]\((x - 2)^6 = (x + (-2))^6\)[/tex] using:
[tex]\[
\sum_{k=0}^{6} \binom{6}{k} x^{6-k} (-2)^k
\][/tex]
Breaking it down:
- For [tex]\(k = 0\)[/tex]: [tex]\(\binom{6}{0} x^{6} (-2)^0 = 1 \cdot x^6 = x^6\)[/tex]
- For [tex]\(k = 1\)[/tex]: [tex]\(\binom{6}{1} x^{5} (-2)^1 = 6 \cdot x^5 \cdot (-2) = -12x^5\)[/tex]
- For [tex]\(k = 2\)[/tex]: [tex]\(\binom{6}{2} x^{4} (-2)^2 = 15 \cdot x^4 \cdot 4 = 60x^4\)[/tex]
- For [tex]\(k = 3\)[/tex]: [tex]\(\binom{6}{3} x^{3} (-2)^3 = 20 \cdot x^3 \cdot (-8) = -160x^3\)[/tex]
- For [tex]\(k = 4\)[/tex]: [tex]\(\binom{6}{4} x^{2} (-2)^4 = 15 \cdot x^2 \cdot 16 = 240x^2\)[/tex]
- For [tex]\(k = 5\)[/tex]: [tex]\(\binom{6}{5} x^{1} (-2)^5 = 6 \cdot x \cdot (-32) = -192x\)[/tex]
- For [tex]\(k = 6\)[/tex]: [tex]\(\binom{6}{6} x^{0} (-2)^6 = 1 \cdot 64 = 64\)[/tex]
4. Combine all terms: Putting it all together, we get
[tex]\[
x^6 - 12x^5 + 60x^4 - 160x^3 + 240x^2 - 192x + 64
\][/tex]
So, the correct expansion of [tex]\((x - 2)^6\)[/tex] is option C:
[tex]\(x^6 - 12x^5 + 60x^4 - 160x^3 + 240x^2 - 192x + 64\)[/tex].
1. Understand 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.
2. Identify variables: In this expression, [tex]\(a = x\)[/tex], [tex]\(b = -2\)[/tex], and [tex]\(n = 6\)[/tex].
3. Apply the Binomial Theorem: We need to expand [tex]\((x - 2)^6 = (x + (-2))^6\)[/tex] using:
[tex]\[
\sum_{k=0}^{6} \binom{6}{k} x^{6-k} (-2)^k
\][/tex]
Breaking it down:
- For [tex]\(k = 0\)[/tex]: [tex]\(\binom{6}{0} x^{6} (-2)^0 = 1 \cdot x^6 = x^6\)[/tex]
- For [tex]\(k = 1\)[/tex]: [tex]\(\binom{6}{1} x^{5} (-2)^1 = 6 \cdot x^5 \cdot (-2) = -12x^5\)[/tex]
- For [tex]\(k = 2\)[/tex]: [tex]\(\binom{6}{2} x^{4} (-2)^2 = 15 \cdot x^4 \cdot 4 = 60x^4\)[/tex]
- For [tex]\(k = 3\)[/tex]: [tex]\(\binom{6}{3} x^{3} (-2)^3 = 20 \cdot x^3 \cdot (-8) = -160x^3\)[/tex]
- For [tex]\(k = 4\)[/tex]: [tex]\(\binom{6}{4} x^{2} (-2)^4 = 15 \cdot x^2 \cdot 16 = 240x^2\)[/tex]
- For [tex]\(k = 5\)[/tex]: [tex]\(\binom{6}{5} x^{1} (-2)^5 = 6 \cdot x \cdot (-32) = -192x\)[/tex]
- For [tex]\(k = 6\)[/tex]: [tex]\(\binom{6}{6} x^{0} (-2)^6 = 1 \cdot 64 = 64\)[/tex]
4. Combine all terms: Putting it all together, we get
[tex]\[
x^6 - 12x^5 + 60x^4 - 160x^3 + 240x^2 - 192x + 64
\][/tex]
So, the correct expansion of [tex]\((x - 2)^6\)[/tex] is option C:
[tex]\(x^6 - 12x^5 + 60x^4 - 160x^3 + 240x^2 - 192x + 64\)[/tex].
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