In the expansion of (2x + 3)^4, what is the coefficient of x^2?

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Question: In the expansion of (2x + 3)^4, what is the coefficient of x^2?

Options:

Correct Answer: 72

Solution:

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

In the expansion of (2x + 3)^4, what is the coefficient of x^2?

In the expansion of (2x + 3)^4, what is the coefficient of x^2?

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 n-k, which is 4-2 = 2, so we need the term with k = 2.
Step 8: Calculate the binomial coefficient C(4, 2), which is the number of ways to choose 2 from 4. C(4, 2) = 4! / (2! * (4-2)!) = 6.
Step 9: Calculate (2)^2, which is 4.
Step 10: Calculate (3)^2, which is 9.
Step 11: Multiply these values together: Coefficient = C(4, 2) * (2)^2 * (3)^2 = 6 * 4 * 9.
Step 12: Finally, calculate 6 * 4 * 9 = 216.

Binomial Expansion – The question tests the understanding of the binomial theorem, specifically how to find coefficients in the expansion of a binomial expression.
Combinatorics – The use of combinations (C(4,2)) to determine how many ways to choose terms from the expansion is a key concept.
Exponentiation – Understanding how to correctly apply exponents to the coefficients in the binomial expansion is crucial.