Find the coefficient of x^1 in the expansion of (x + 2)^5.

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Question: Find the coefficient of x^1 in the expansion of (x + 2)^5.

Options:

Correct Answer: 20

Solution:

The coefficient of x^1 is C(5,1) * 2^4 = 5 * 16 = 80.

Find the coefficient of x^1 in the expansion of (x + 2)^5.

Find the coefficient of x^1 in the expansion of (x + 2)^5.

Correct Answer: 80

Step 1: Identify the expression to expand, which is (x + 2)^5.
Step 2: Understand that we want the coefficient of x^1 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 = 2, and n = 5.
Step 5: We need to find the term where x is raised to the power of 1, which means k = 4 (since n - k = 1).
Step 6: Calculate C(5, 1), which is the number of ways to choose 1 from 5. This is equal to 5.
Step 7: Calculate 2^4, which is 16.
Step 8: Multiply the results from Step 6 and Step 7: 5 * 16 = 80.
Step 9: Conclude that the coefficient of x^1 in the expansion of (x + 2)^5 is 80.

Binomial Expansion – The process of expanding expressions of the form (a + b)^n using the binomial theorem.
Coefficients in Binomial Expansion – Understanding how to find specific coefficients in the expansion using combinations.