Calculate the coefficient of x^2 in the expansion of (2x + 3)^4.

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Question: Calculate the coefficient of x^2 in the expansion of (2x + 3)^4.

Options:

Correct Answer: 48

Solution:

The coefficient of x^2 is C(4,2) * (2)^2 * (3)^2 = 6 * 4 * 9 = 216.

Calculate the coefficient of x^2 in the expansion of (2x + 3)^4.

Calculate the coefficient of x^2 in the expansion of (2x + 3)^4.

Step 1: Identify the expression to expand, which is (2x + 3)^4.
Step 2: Recognize that we need 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 = 2x, b = 3, and n = 4.
Step 5: We want the term where x has the power of 2, which means we need to find the term where (2x) is raised to the power of 2.
Step 6: This corresponds to k = 2 in the Binomial Theorem, since we want (2x)^(n-k) = (2x)^2.
Step 7: Calculate C(4, 2), which is the number of ways to choose 2 from 4. C(4, 2) = 4! / (2! * (4-2)!) = 6.
Step 8: Calculate (2)^2, which is 4.
Step 9: Calculate (3)^2, which is 9.
Step 10: Multiply these values together: Coefficient = C(4, 2) * (2)^2 * (3)^2 = 6 * 4 * 9.
Step 11: Perform the multiplication: 6 * 4 = 24, then 24 * 9 = 216.
Step 12: Conclude that the coefficient of x^2 in the expansion of (2x + 3)^4 is 216.

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 and powers.