What is the coefficient of x^7 in the expansion of (2x - 3)^8? (2022)

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Question: What is the coefficient of x^7 in the expansion of (2x - 3)^8? (2022)

Options:

Correct Answer: -6720

Solution:

The coefficient of x^7 is C(8,7)(2)^7(-3)^1 = 8 * 128 * -3 = -3072.

What is the coefficient of x^7 in the expansion of (2x - 3)^8? (2022)

What is the coefficient of x^7 in the expansion of (2x - 3)^8? (2022)

Step 1: Identify the expression we need to expand, which is (2x - 3)^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 = -3, and n = 8.
Step 4: We want the coefficient of x^7, which means we need the term where (2x) is raised to the power of 7.
Step 5: This occurs when k = 1 because (2x)^(8-k) = (2x)^7 when k = 1.
Step 6: Calculate C(8, 7), which is the number of ways to choose 7 items from 8. This equals 8.
Step 7: Calculate (2)^7, which is 128.
Step 8: Calculate (-3)^1, which is -3.
Step 9: Multiply these values together: 8 (from C(8, 7)) * 128 (from (2)^7) * -3 (from (-3)^1).
Step 10: The final calculation is 8 * 128 * -3 = -3072.

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