In the expansion of (x - 1)^8, what is the coefficient of x^5?

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Question: In the expansion of (x - 1)^8, what is the coefficient of x^5?

Options:

Correct Answer: -56

Solution:

The coefficient of x^5 in (x - 1)^8 is C(8,5) * (-1)^3 = 56 * (-1) = -56.

In the expansion of (x - 1)^8, what is the coefficient of x^5?

In the expansion of (x - 1)^8, what is the coefficient of x^5?

Step 1: Identify the expression we are working with, which is (x - 1)^8.
Step 2: Recognize that we need to find the coefficient of x^5 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 = -1, and n = 8.
Step 5: We want the term where x is raised to the power of 5, which means we need to find the term where k = 3 (since 8 - 5 = 3).
Step 6: Calculate the binomial coefficient C(8, 3), which is the number of ways to choose 3 items from 8.
Step 7: C(8, 3) = 8! / (3! * (8-3)!) = 56.
Step 8: The term we are interested in is C(8, 3) * (x)^(8-3) * (-1)^3.
Step 9: Substitute the values: 56 * x^5 * (-1)^3.
Step 10: Calculate (-1)^3, which is -1.
Step 11: Multiply the coefficient: 56 * (-1) = -56.
Step 12: Conclude that the coefficient of x^5 in the expansion of (x - 1)^8 is -56.

Binomial Expansion – The question tests understanding of the binomial theorem, specifically how to find coefficients in the expansion of a binomial expression.
Combinatorics – The use of combinations (C(n, k)) to determine the number of ways to choose k successes in n trials is essential for finding the coefficient.
Sign of Terms – Understanding how the sign of terms changes based on the powers of negative numbers in the expansion.