What is the coefficient of x^2 in the expansion of (x - 1)^5?

₹0.0

Question: What is the coefficient of x^2 in the expansion of (x - 1)^5?

Options:

Correct Answer: -10

Solution:

The coefficient of x^2 is C(5,2) * (-1)^3 = 10 * (-1) = -10.

What is the coefficient of x^2 in the expansion of (x - 1)^5?

What is the coefficient of x^2 in the expansion of (x - 1)^5?

Step 1: Identify the expression we need to expand, which is (x - 1)^5.
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 = -1, and n = 5.
Step 4: We want the coefficient of x^2, which corresponds to k = 3 (since n - k = 2).
Step 5: Calculate C(5, 3), which is the number of ways to choose 3 items from 5. This is equal to 5! / (3! * (5-3)!) = 10.
Step 6: Since b = -1, we need to include (-1)^3 in our calculation, which is -1.
Step 7: Multiply the coefficient C(5, 3) by (-1)^3: 10 * (-1) = -10.
Step 8: Therefore, the coefficient of x^2 in the expansion of (x - 1)^5 is -10.

Binomial Expansion – The process of expanding expressions of the form (a + b)^n using the binomial theorem.
Binomial Coefficient – The coefficient C(n, k) represents the number of ways to choose k elements from a set of n elements.
Negative Exponents – Understanding how to handle negative terms in the expansion, particularly when using the binomial theorem.