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

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

Options:

Correct Answer: -40

Solution:

The coefficient of x^2 is given by 5C2 * (-2)^3 = 10 * (-8) = -80.

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

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

Step 1: Identify the expression to expand, which is (x - 2)^5.
Step 2: Use the Binomial Theorem, which states that (a + b)^n = sum of (nCk * a^(n-k) * b^k) for k from 0 to n.
Step 3: In our case, a = x, b = -2, and n = 5.
Step 4: We want the coefficient of x^2, which corresponds to k = 3 (since n - k = 2).
Step 5: Calculate nCk, which is 5C3. This is equal to 5! / (3! * (5-3)!) = 10.
Step 6: Calculate (-2)^3, which is -2 * -2 * -2 = -8.
Step 7: Multiply the results from Step 5 and Step 6: 10 * (-8) = -80.
Step 8: Conclude that the coefficient of x^2 in the expansion of (x - 2)^5 is -80.

Binomial Expansion – The process of expanding expressions of the form (a + b)^n using the binomial theorem.
Binomial Coefficient – The coefficient in the expansion, represented as nCk, which gives the number of ways to choose k elements from a set of n elements.
Negative Exponents – Understanding how to handle negative numbers in the expansion, particularly when raised to a power.