What is the coefficient of x^0 in the expansion of (x - 1)^5?

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Question: What is the coefficient of x^0 in the expansion of (x - 1)^5?

Options:

Correct Answer: -5

Solution:

The coefficient of x^0 in (x - 1)^5 is given by 5C5 * (-1)^5 = -1.

What is the coefficient of x^0 in the expansion of (x - 1)^5?

What is the coefficient of x^0 in the expansion of (x - 1)^5?

Step 1: Understand that x^0 means we are looking for the constant term in the expansion of (x - 1)^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 = -1, and n = 5. We need to find the term where x^0 appears.
Step 4: The term with x^0 occurs when k = 5 (because we want all x's to cancel out).
Step 5: Calculate the coefficient for k = 5: This is given by 5C5 * (x)^(5-5) * (-1)^5.
Step 6: Calculate 5C5, which is 1 (since there is only one way to choose all 5 items).
Step 7: Calculate (-1)^5, which is -1.
Step 8: Multiply the results: 1 * -1 = -1.
Step 9: Conclude that the coefficient of x^0 in the expansion of (x - 1)^5 is -1.

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