What is the coefficient of x^3 in the expansion of (x - 2)^6?

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Question: What is the coefficient of x^3 in the expansion of (x - 2)^6?

Options:

Correct Answer: -120

Solution:

The coefficient of x^3 is C(6,3) * (-2)^3 = 20 * (-8) = -160.

What is the coefficient of x^3 in the expansion of (x - 2)^6?

What is the coefficient of x^3 in the expansion of (x - 2)^6?

Step 1: Identify the expression we need to expand, which is (x - 2)^6.
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 = x, b = -2, and n = 6.
Step 4: We want the coefficient of x^3, which means we need to find the term where the power of x is 3.
Step 5: To find this term, we set n-k = 3, which means k = 6 - 3 = 3.
Step 6: Calculate C(6, 3), which is the number of ways to choose 3 items from 6. This is equal to 6! / (3! * (6-3)!) = 20.
Step 7: Now calculate (-2)^3, which is -2 * -2 * -2 = -8.
Step 8: Multiply the coefficient C(6, 3) by (-2)^3: 20 * (-8) = -160.
Step 9: The coefficient of x^3 in the expansion of (x - 2)^6 is -160.

Binomial Expansion – The question tests the understanding of the binomial theorem, specifically how to find coefficients in the expansion of a binomial expression.
Combinatorics – It involves calculating combinations, represented as C(n, k), which is crucial for determining the coefficients in the expansion.
Negative Exponents – The question requires handling negative numbers, as the term (-2) raised to a power is involved in the calculation.