In the expansion of (2x - 5)^5, what is the coefficient of x^3? (2021)

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Question: In the expansion of (2x - 5)^5, what is the coefficient of x^3? (2021)

Options:

Correct Answer: 450

Exam Year: 2021

Solution:

The coefficient of x^3 is C(5,3) * (2)^3 * (-5)^2 = 10 * 8 * 25 = 2000.

In the expansion of (2x - 5)^5, what is the coefficient of x^3? (2021)

In the expansion of (2x - 5)^5, what is the coefficient of x^3? (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^3 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 = 5.
Step 5: We want the term where x has the power of 3, which means we need to find the term where (2x) is raised to the power of 3.
Step 6: This corresponds to k = 2 in the Binomial Theorem because n - k = 3 (5 - k = 3). So, k = 2.
Step 7: Calculate C(5, 2), which is the number of ways to choose 2 from 5. C(5, 2) = 5! / (2!(5-2)!) = 10.
Step 8: Calculate (2)^3, which is 2 * 2 * 2 = 8.
Step 9: Calculate (-5)^2, which is (-5) * (-5) = 25.
Step 10: Multiply these values together: Coefficient = C(5, 2) * (2)^3 * (-5)^2 = 10 * 8 * 25.
Step 11: Calculate the final result: 10 * 8 = 80, and then 80 * 25 = 2000.
Step 12: Therefore, the coefficient of x^3 in the expansion of (2x - 5)^5 is 2000.

Binomial Expansion – The question tests the understanding of the binomial theorem, specifically how to find coefficients in the expansion of a binomial expression.
Combinatorics – The use of combinations (C(n, k)) to determine the number of ways to choose terms from the expansion.
Exponent Rules – Applying exponent rules to calculate powers of the coefficients in the binomial expansion.