In the expansion of (2x - 5)^5, what is the coefficient of x^2? (2021)

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Question: In the expansion of (2x - 5)^5, what is the coefficient of x^2? (2021)

Options:

Correct Answer: -300

Exam Year: 2021

Solution:

The coefficient of x^2 is C(5,2) * (2)^2 * (-5)^3 = 10 * 4 * (-125) = -5000.

In the expansion of (2x - 5)^5, what is the coefficient of x^2? (2021)

In the expansion of (2x - 5)^5, what is the coefficient of x^2? (2021)

Step 1: Identify the expression to expand, which is (2x - 5)^5.
Step 2: Recognize that we need to find the coefficient of x^2 in the expansion.
Step 3: 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 4: In our case, a = 2x, b = -5, and n = 5.
Step 5: We want the term where x has the power of 2, which means we need to find the term where (2x) is raised to the power of 2.
Step 6: This corresponds to k = 3 because n - k = 2 (5 - k = 2). So, k = 3.
Step 7: Calculate C(5, 3), which is the number of ways to choose 3 items from 5. C(5, 3) = 5! / (3! * (5-3)!) = 10.
Step 8: Calculate (2)^2, which is 4.
Step 9: Calculate (-5)^3, which is -125.
Step 10: Multiply these values together: Coefficient = C(5, 3) * (2)^2 * (-5)^3 = 10 * 4 * (-125).
Step 11: Perform the multiplication: 10 * 4 = 40, then 40 * (-125) = -5000.
Step 12: Conclude that the coefficient of x^2 in the expansion of (2x - 5)^5 is -5000.

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