Answer :
To expand [tex]\((x-2)^6\)[/tex] using the Binomial Theorem, we follow these steps:
The Binomial 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], we can set it up as:
- [tex]\(a = x\)[/tex]
- [tex]\(b = -2\)[/tex]
- [tex]\(n = 6\)[/tex]
The expansion will look like this:
[tex]\[
(x - 2)^6 = \sum_{k=0}^{6} \binom{6}{k} x^{6-k} (-2)^k
\][/tex]
Now, let's calculate each term in the expansion:
1. Term [tex]\(k=0\)[/tex]:
- [tex]\(\binom{6}{0} = 1\)[/tex]
- The term is: [tex]\(1 \cdot x^6 \cdot (-2)^0 = x^6\)[/tex]
2. Term [tex]\(k=1\)[/tex]:
- [tex]\(\binom{6}{1} = 6\)[/tex]
- The term is: [tex]\(6 \cdot x^5 \cdot (-2)^1 = -12x^5\)[/tex]
3. Term [tex]\(k=2\)[/tex]:
- [tex]\(\binom{6}{2} = 15\)[/tex]
- The term is: [tex]\(15 \cdot x^4 \cdot (-2)^2 = 60x^4\)[/tex]
4. Term [tex]\(k=3\)[/tex]:
- [tex]\(\binom{6}{3} = 20\)[/tex]
- The term is: [tex]\(20 \cdot x^3 \cdot (-2)^3 = -160x^3\)[/tex]
5. Term [tex]\(k=4\)[/tex]:
- [tex]\(\binom{6}{4} = 15\)[/tex]
- The term is: [tex]\(15 \cdot x^2 \cdot (-2)^4 = 240x^2\)[/tex]
6. Term [tex]\(k=5\)[/tex]:
- [tex]\(\binom{6}{5} = 6\)[/tex]
- The term is: [tex]\(6 \cdot x^1 \cdot (-2)^5 = -192x\)[/tex]
7. Term [tex]\(k=6\)[/tex]:
- [tex]\(\binom{6}{6} = 1\)[/tex]
- The term is: [tex]\(1 \cdot x^0 \cdot (-2)^6 = 64\)[/tex]
Putting it all together, the expansion of [tex]\((x - 2)^6\)[/tex] is:
[tex]\[
x^6 - 12x^5 + 60x^4 - 160x^3 + 240x^2 - 192x + 64
\][/tex]
Thus, the correct option is C:
[tex]\[x^6-12 x^5+60 x^4-160 x^3+240 x^2-192 x+64\][/tex]
The Binomial 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], we can set it up as:
- [tex]\(a = x\)[/tex]
- [tex]\(b = -2\)[/tex]
- [tex]\(n = 6\)[/tex]
The expansion will look like this:
[tex]\[
(x - 2)^6 = \sum_{k=0}^{6} \binom{6}{k} x^{6-k} (-2)^k
\][/tex]
Now, let's calculate each term in the expansion:
1. Term [tex]\(k=0\)[/tex]:
- [tex]\(\binom{6}{0} = 1\)[/tex]
- The term is: [tex]\(1 \cdot x^6 \cdot (-2)^0 = x^6\)[/tex]
2. Term [tex]\(k=1\)[/tex]:
- [tex]\(\binom{6}{1} = 6\)[/tex]
- The term is: [tex]\(6 \cdot x^5 \cdot (-2)^1 = -12x^5\)[/tex]
3. Term [tex]\(k=2\)[/tex]:
- [tex]\(\binom{6}{2} = 15\)[/tex]
- The term is: [tex]\(15 \cdot x^4 \cdot (-2)^2 = 60x^4\)[/tex]
4. Term [tex]\(k=3\)[/tex]:
- [tex]\(\binom{6}{3} = 20\)[/tex]
- The term is: [tex]\(20 \cdot x^3 \cdot (-2)^3 = -160x^3\)[/tex]
5. Term [tex]\(k=4\)[/tex]:
- [tex]\(\binom{6}{4} = 15\)[/tex]
- The term is: [tex]\(15 \cdot x^2 \cdot (-2)^4 = 240x^2\)[/tex]
6. Term [tex]\(k=5\)[/tex]:
- [tex]\(\binom{6}{5} = 6\)[/tex]
- The term is: [tex]\(6 \cdot x^1 \cdot (-2)^5 = -192x\)[/tex]
7. Term [tex]\(k=6\)[/tex]:
- [tex]\(\binom{6}{6} = 1\)[/tex]
- The term is: [tex]\(1 \cdot x^0 \cdot (-2)^6 = 64\)[/tex]
Putting it all together, the expansion of [tex]\((x - 2)^6\)[/tex] is:
[tex]\[
x^6 - 12x^5 + 60x^4 - 160x^3 + 240x^2 - 192x + 64
\][/tex]
Thus, the correct option is C:
[tex]\[x^6-12 x^5+60 x^4-160 x^3+240 x^2-192 x+64\][/tex]
