Find the coefficient of x^2 in the expansion of (x + 4)^5. (2023)

Find the coefficient of x^2 in the expansion of (x + 4)^5. (2023)

Step 1: Identify the expression we need to expand, which is (x + 4)^5.
Step 2: Understand that we want to find the coefficient of x^2 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 = 5.
Step 5: We need to find the term where x is raised to the power of 2, which means we need k = 3 (since n - k = 2).
Step 6: Calculate C(5, 2), which is the number of ways to choose 2 items from 5. This is equal to 5! / (2! * (5-2)!) = 10.
Step 7: Calculate 4^3, which is 4 * 4 * 4 = 64.
Step 8: Multiply the results from Step 6 and Step 7: 10 * 64 = 640.
Step 9: Conclude that the coefficient of x^2 in the expansion of (x + 4)^5 is 640.

Binomial Expansion – The process of expanding expressions of the form (a + b)^n using the binomial theorem, which involves combinations and powers.
Coefficients in Polynomial Expansion – Identifying and calculating specific coefficients in the expanded form of a polynomial expression.