Find the coefficient of x^3 in the expansion of (x - 2)^5.

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Question: Find the coefficient of x^3 in the expansion of (x - 2)^5.

Options:

Correct Answer: -40

Solution:

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

Find the coefficient of x^3 in the expansion of (x - 2)^5.

Find 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: Recognize that we want 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 = x, b = -2, and n = 5.
Step 5: We need to find the term where x is raised to the power of 3, which means we need k = 2 (since n - k = 3).
Step 6: Calculate C(5, 2), which is the number of ways to choose 2 from 5. C(5, 2) = 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 process of expanding expressions of the form (a + b)^n using the binomial theorem.
Coefficients in Binomial Expansion – Understanding how to find specific coefficients in the expansion using combinations.