What is the coefficient of x^6 in the expansion of (2x - 1)^8? (2022)

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Question: What is the coefficient of x^6 in the expansion of (2x - 1)^8? (2022)

Options:

Correct Answer: -112

Exam Year: 2022

Solution:

The coefficient of x^6 is C(8,6) * (2)^6 * (-1)^2 = 28 * 64 = 1792.

What is the coefficient of x^6 in the expansion of (2x - 1)^8? (2022)

What is the coefficient of x^6 in the expansion of (2x - 1)^8? (2022)

Step 1: Identify the expression we need to expand, which is (2x - 1)^8.
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 = 2x, b = -1, and n = 8.
Step 4: We want the term that contains x^6. This occurs when (2x) is raised to the power of 6.
Step 5: To find the corresponding k, we set n - k = 6, which gives us k = 8 - 6 = 2.
Step 6: Calculate the binomial coefficient C(8, 2), which is the number of ways to choose 2 from 8.
Step 7: C(8, 2) = 8! / (2! * (8-2)!) = 28.
Step 8: Now calculate (2x)^6, which is (2^6) * (x^6) = 64 * x^6.
Step 9: Next, calculate (-1)^2, which is 1.
Step 10: Combine these results: Coefficient = C(8, 2) * (2^6) * (-1)^2 = 28 * 64 * 1.
Step 11: Finally, calculate 28 * 64 = 1792.

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