In the expansion of (2x - 5)^6, what is the coefficient of x^3? (2020)

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Question: In the expansion of (2x - 5)^6, what is the coefficient of x^3? (2020)

Options:

Correct Answer: -720

Solution:

The coefficient of x^3 is C(6,3)(2)^3(-5)^3 = 20 * 8 * -125 = -20000.

In the expansion of (2x - 5)^6, what is the coefficient of x^3? (2020)

In the expansion of (2x - 5)^6, what is the coefficient of x^3? (2020)

Step 1: Identify the expression to expand, which is (2x - 5)^6.
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 = 6.
Step 5: We want the term where x has the power of 3, which means we need to find the term where k = 3 (since x^3 comes from (2x)^3).
Step 6: Calculate C(6, 3), which is the number of ways to choose 3 items from 6. C(6, 3) = 6! / (3! * (6-3)!) = 20.
Step 7: Calculate (2)^3, which is 2 * 2 * 2 = 8.
Step 8: Calculate (-5)^3, which is -5 * -5 * -5 = -125.
Step 9: Combine these values to find the coefficient: C(6, 3) * (2)^3 * (-5)^3 = 20 * 8 * -125.
Step 10: Perform the multiplication: 20 * 8 = 160, and then 160 * -125 = -20000.
Step 11: Conclude that the coefficient of x^3 in the expansion of (2x - 5)^6 is -20000.

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 terms in the binomial expression.