In the expansion of (2x - 5)^6, what is the coefficient of x^2? (2021)

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Question: In the expansion of (2x - 5)^6, what is the coefficient of x^2? (2021)

Options:

Correct Answer: -150

Exam Year: 2021

Solution:

The coefficient of x^2 is C(6,2) * (2)^2 * (-5)^4 = 15 * 4 * 625 = -37500.

In the expansion of (2x - 5)^6, what is the coefficient of x^2? (2021)

In the expansion of (2x - 5)^6, what is the coefficient of x^2? (2021)

Step 1: Identify the expression to expand, which is (2x - 5)^6.
Step 2: Recognize that we need to find the coefficient of x^2 in this 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 = 6.
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 = 4 because n - k = 2 (6 - k = 2). So, k = 4.
Step 7: Calculate C(6, 4), which is the number of ways to choose 4 from 6. C(6, 4) = 15.
Step 8: Calculate (2)^2, which is 4.
Step 9: Calculate (-5)^4, which is 625.
Step 10: Multiply these values together: Coefficient = C(6, 4) * (2)^2 * (-5)^4 = 15 * 4 * 625.
Step 11: Perform the multiplication: 15 * 4 = 60, and then 60 * 625 = 37500.
Step 12: Since we are considering (-5)^4, the coefficient is positive, so the final answer is 37500.

Binomial Expansion – The process of expanding expressions of the form (a + b)^n using the Binomial Theorem.
Coefficients in Binomial Expansion – Understanding how to calculate specific coefficients in the expansion using combinations and powers.