Find the coefficient of x^1 in the expansion of (x + 4)^3.

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Question: Find the coefficient of x^1 in the expansion of (x + 4)^3.

Options:

Correct Answer: 12

Solution:

The coefficient of x^1 is C(3,1) * (4)^2 = 3 * 16 = 48.

Find the coefficient of x^1 in the expansion of (x + 4)^3.

Find the coefficient of x^1 in the expansion of (x + 4)^3.

Step 1: Identify the expression we need to expand, which is (x + 4)^3.
Step 2: Understand that we want to find 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 = 4, and n = 3.
Step 5: We need to find the term where the power of x is 1, which means we want k = 2 (since n - k = 1).
Step 6: Calculate C(3, 2), which is the number of ways to choose 2 from 3. C(3, 2) = 3.
Step 7: Calculate 4^2, which is 16.
Step 8: Multiply the results from Step 6 and Step 7: 3 * 16 = 48.
Step 9: Conclude that the coefficient of x^1 in the expansion of (x + 4)^3 is 48.

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