What is the 5th term in the expansion of (x - 1)^8? (2022)

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Question: What is the 5th term in the expansion of (x - 1)^8? (2022)

Options:

Correct Answer: -56x^4

Exam Year: 2022

Solution:

The 5th term corresponds to k=4. Using the binomial theorem, it is given by 8C4 * (x)^4 * (-1)^4 = 70x^4.

What is the 5th term in the expansion of (x - 1)^8? (2022)

What is the 5th term in the expansion of (x - 1)^8? (2022)

Step 1: Identify the expression we are expanding, which is (x - 1)^8.
Step 2: Understand that we need to find the 5th term in the expansion.
Step 3: Recall that in the binomial expansion, the k-th term can be found using the formula: nCk * (a)^(n-k) * (b)^k, where n is the exponent, a is the first term, b is the second term, and k starts from 0.
Step 4: For our expression (x - 1)^8, we have n = 8, a = x, and b = -1.
Step 5: The 5th term corresponds to k = 4 (since we start counting from k = 0).
Step 6: Calculate the binomial coefficient: 8C4, which is the number of ways to choose 4 items from 8. This equals 70.
Step 7: Calculate (x)^(8-4) = (x)^4.
Step 8: Calculate (-1)^4, which equals 1.
Step 9: Combine all parts: 8C4 * (x)^4 * (-1)^4 = 70 * (x)^4 * 1 = 70x^4.
Step 10: Therefore, the 5th term in the expansion of (x - 1)^8 is 70x^4.

Binomial Expansion – The process of expanding expressions of the form (a + b)^n using the binomial theorem, which involves combinations and powers.
Binomial Coefficient – The coefficient in the binomial expansion, represented as nCk, which calculates the number of ways to choose k elements from a set of n elements.
Term Identification – Understanding how to identify specific terms in a binomial expansion based on the index k.