Answer :
To expand [tex]\((x - 2)^6\)[/tex] using the Binomial Theorem, we use the formula:
[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 expansion is given by summing up the terms:
[tex]\[
\sum_{k=0}^{6} \binom{6}{k} x^{6-k} (-2)^k
\][/tex]
Let's go through the expansion step-by-step:
1. [tex]\(k = 0\)[/tex]:
[tex]\(\binom{6}{0} x^{6-0} (-2)^0 = 1 \cdot x^6 \cdot 1 = x^6 \)[/tex]
2. [tex]\(k = 1\)[/tex]:
[tex]\(\binom{6}{1} x^{6-1} (-2)^1 = 6 \cdot x^5 \cdot (-2) = -12x^5 \)[/tex]
3. [tex]\(k = 2\)[/tex]:
[tex]\(\binom{6}{2} x^{6-2} (-2)^2 = 15 \cdot x^4 \cdot 4 = 60x^4 \)[/tex]
4. [tex]\(k = 3\)[/tex]:
[tex]\(\binom{6}{3} x^{6-3} (-2)^3 = 20 \cdot x^3 \cdot (-8) = -160x^3 \)[/tex]
5. [tex]\(k = 4\)[/tex]:
[tex]\(\binom{6}{4} x^{6-4} (-2)^4 = 15 \cdot x^2 \cdot 16 = 240x^2 \)[/tex]
6. [tex]\(k = 5\)[/tex]:
[tex]\(\binom{6}{5} x^{6-5} (-2)^5 = 6 \cdot x^1 \cdot (-32) = -192x \)[/tex]
7. [tex]\(k = 6\)[/tex]:
[tex]\(\binom{6}{6} x^{6-6} (-2)^6 = 1 \cdot x^0 \cdot 64 = 64 \)[/tex]
Adding these terms together gives the expanded form of [tex]\((x - 2)^6\)[/tex]:
[tex]\[ x^6 - 12x^5 + 60x^4 - 160x^3 + 240x^2 - 192x + 64 \][/tex]
Therefore, the correct answer is:
C. [tex]\(x^6 - 12x^5 + 60x^4 - 160x^3 + 240x^2 - 192x + 64\)[/tex]
[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 expansion is given by summing up the terms:
[tex]\[
\sum_{k=0}^{6} \binom{6}{k} x^{6-k} (-2)^k
\][/tex]
Let's go through the expansion step-by-step:
1. [tex]\(k = 0\)[/tex]:
[tex]\(\binom{6}{0} x^{6-0} (-2)^0 = 1 \cdot x^6 \cdot 1 = x^6 \)[/tex]
2. [tex]\(k = 1\)[/tex]:
[tex]\(\binom{6}{1} x^{6-1} (-2)^1 = 6 \cdot x^5 \cdot (-2) = -12x^5 \)[/tex]
3. [tex]\(k = 2\)[/tex]:
[tex]\(\binom{6}{2} x^{6-2} (-2)^2 = 15 \cdot x^4 \cdot 4 = 60x^4 \)[/tex]
4. [tex]\(k = 3\)[/tex]:
[tex]\(\binom{6}{3} x^{6-3} (-2)^3 = 20 \cdot x^3 \cdot (-8) = -160x^3 \)[/tex]
5. [tex]\(k = 4\)[/tex]:
[tex]\(\binom{6}{4} x^{6-4} (-2)^4 = 15 \cdot x^2 \cdot 16 = 240x^2 \)[/tex]
6. [tex]\(k = 5\)[/tex]:
[tex]\(\binom{6}{5} x^{6-5} (-2)^5 = 6 \cdot x^1 \cdot (-32) = -192x \)[/tex]
7. [tex]\(k = 6\)[/tex]:
[tex]\(\binom{6}{6} x^{6-6} (-2)^6 = 1 \cdot x^0 \cdot 64 = 64 \)[/tex]
Adding these terms together gives the expanded form of [tex]\((x - 2)^6\)[/tex]:
[tex]\[ x^6 - 12x^5 + 60x^4 - 160x^3 + 240x^2 - 192x + 64 \][/tex]
Therefore, the correct answer is:
C. [tex]\(x^6 - 12x^5 + 60x^4 - 160x^3 + 240x^2 - 192x + 64\)[/tex]
