What is the coefficient of x^3 in the expansion of (x - 2)^5?

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Question: What is the coefficient of x^3 in the expansion of (x - 2)^5?

Options:

Correct Answer: 10

Solution:

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

What is the coefficient of x^3 in the expansion of (x - 2)^5?

What is the coefficient of x^3 in the expansion of (x - 2)^5?

Step 1: Identify the expression we need to expand, which is (x - 2)^5.
Step 2: Understand that we want 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 = x, b = -2, and n = 5.
Step 5: We need to find the term where the power of x is 3, which means we need k = 2 (since 5 - k = 3).
Step 6: Calculate C(5, 2), which is the number of ways to choose 2 from 5. This is equal to 5! / (2! * (5-2)!) = 10.
Step 7: Calculate (-2)^2, which is 4.
Step 8: Multiply the results from Step 6 and Step 7: 10 * 4 = 40.
Step 9: Conclude that the coefficient of x^3 in the expansion of (x - 2)^5 is 40.

Binomial Expansion – The question tests the understanding of the binomial theorem, specifically how to find specific coefficients in the expansion of a binomial expression.
Combinatorial Coefficients – It requires knowledge of how to calculate combinations, represented as C(n, k), which is essential for determining the coefficients in the expansion.
Negative Exponents – The problem involves handling negative constants in the binomial expression, which can affect the sign of the resulting coefficients.