What is the coefficient of x^0 in the expansion of (x + 1/x)^8? (2021)

What is the coefficient of x^0 in the expansion of (x + 1/x)^8? (2021)

Step 1: Understand that we need to find the coefficient of x^0 in the expression (x + 1/x)^8.
Step 2: Recognize that x^0 means we want the term where x is not present, which happens when the powers of x cancel out.
Step 3: Use the binomial expansion formula: (a + b)^n = Σ (nCk) * a^(n-k) * b^k, where n is the total power, a is the first term, b is the second term, and k is the term number.
Step 4: In our case, a = x, b = 1/x, and n = 8.
Step 5: We need to find k such that the power of x in the term is 0. The general term in the expansion is (8Ck) * (x)^(8-k) * (1/x)^k.
Step 6: Simplify the term: (8Ck) * (x)^(8-k) * (x^(-k)) = (8Ck) * (x)^(8-k-k) = (8Ck) * (x)^(8-2k).
Step 7: Set the exponent of x to 0: 8 - 2k = 0.
Step 8: Solve for k: 8 = 2k, so k = 4.
Step 9: Now, find the coefficient by calculating 8C4, which is the number of ways to choose 4 items from 8.
Step 10: Calculate 8C4 = 8! / (4! * (8-4)!) = 70.
Step 11: Conclude that the coefficient of x^0 in the expansion of (x + 1/x)^8 is 70.

No concepts available.