What is the coefficient of x^0 in the expansion of (2x - 3)^5? (2022)

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Question: What is the coefficient of x^0 in the expansion of (2x - 3)^5? (2022)

Options:

Correct Answer: -243

Solution:

The constant term is C(5,5)(-3)^5 = -243.

What is the coefficient of x^0 in the expansion of (2x - 3)^5? (2022)

What is the coefficient of x^0 in the expansion of (2x - 3)^5? (2022)

Step 1: Understand that x^0 means we are looking for the constant term in the expansion of (2x - 3)^5.
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 = 5.
Step 4: To find the constant term, we need to set the power of x to 0. This happens when we choose all of b (-3) and none of a (2x).
Step 5: The term we need is when k = 5 (which means we take (-3) five times).
Step 6: Calculate the coefficient using C(5, 5) which is 1, and (-3)^5 which is -243.
Step 7: Multiply the coefficient by (-3)^5: 1 * (-243) = -243.
Step 8: Therefore, the coefficient of x^0 (the constant term) in the expansion is -243.

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