Determine the coefficient of x^2 in the expansion of (x - 2)^6.

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Question: Determine the coefficient of x^2 in the expansion of (x - 2)^6.

Options:

Correct Answer: -60

Solution:

The coefficient of x^2 is C(6,2)(-2)^4 = 15 * 16 = 240.

Determine the coefficient of x^2 in the expansion of (x - 2)^6.

Determine the coefficient of x^2 in the expansion of (x - 2)^6.

Correct Answer: 240

Step 1: Identify the expression to expand, which is (x - 2)^6.
Step 2: Use the Binomial Theorem, which states that (a + b)^n = Σ (C(n, k) * a^(n-k) * b^k) for k = 0 to n.
Step 3: In our case, a = x, b = -2, and n = 6.
Step 4: We want the coefficient of x^2, which corresponds to k = 4 (since n - k = 2).
Step 5: Calculate C(6, 4), which is the number of ways to choose 4 items from 6. This is equal to C(6, 2) because C(n, k) = C(n, n-k).
Step 6: C(6, 2) = 6! / (2! * (6-2)!) = 6! / (2! * 4!) = (6 * 5) / (2 * 1) = 15.
Step 7: Calculate (-2)^4, which is 16.
Step 8: Multiply the results from Step 6 and Step 7: 15 * 16 = 240.
Step 9: The coefficient of x^2 in the expansion of (x - 2)^6 is 240.

Binomial Expansion – The process of expanding expressions of the form (a + b)^n using the binomial theorem.
Coefficients in Binomial Expansion – Understanding how to find specific coefficients in the expansion using combinations and powers.