What is the coefficient of x^2 in the expansion of (2x - 1)^5? (2021)

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Question: What is the coefficient of x^2 in the expansion of (2x - 1)^5? (2021)

Options:

Correct Answer: 20

Exam Year: 2021

Solution:

The coefficient of x^2 is C(5,2) * (2)^2 * (-1)^3 = 10 * 4 * (-1) = -40.

What is the coefficient of x^2 in the expansion of (2x - 1)^5? (2021)

What is the coefficient of x^2 in the expansion of (2x - 1)^5? (2021)

Step 1: Identify the expression to expand, which is (2x - 1)^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 = -1, and n = 5.
Step 4: We want the coefficient of x^2, which means we need to find the term where (2x) is raised to the power of 2.
Step 5: This occurs when k = 3 because (n - k) = 2 (since 5 - 3 = 2).
Step 6: Calculate C(5, 2), which is the number of ways to choose 2 items from 5. C(5, 2) = 5! / (2! * (5-2)!) = 10.
Step 7: Calculate (2)^2, which is 4.
Step 8: Calculate (-1)^3, which is -1.
Step 9: Multiply these values together: Coefficient = C(5, 2) * (2)^2 * (-1)^3 = 10 * 4 * (-1).
Step 10: The final result is -40, which is the coefficient of x^2.

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