What is the coefficient of x^4 in the expansion of (2x + 3)^6?

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Question: What is the coefficient of x^4 in the expansion of (2x + 3)^6?

Options:

Correct Answer: 810

Solution:

The coefficient of x^4 is C(6,4) * (2)^4 * (3)^(6-4) = 15 * 16 * 9 = 2160.

What is the coefficient of x^4 in the expansion of (2x + 3)^6?

What is the coefficient of x^4 in the expansion of (2x + 3)^6?

Step 1: Identify the expression to expand, which is (2x + 3)^6.
Step 2: Recognize that we need to find the coefficient of x^4 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 = 6.
Step 5: We want the term where x has the power of 4, which means we need (2x)^4 and (3)^(6-4).
Step 6: Set k = 4, so we need to calculate C(6, 4), (2)^4, and (3)^(6-4).
Step 7: Calculate C(6, 4), which is the number of ways to choose 4 items from 6. This equals 15.
Step 8: Calculate (2)^4, which is 16.
Step 9: Calculate (3)^(6-4), which is (3)^2, and that equals 9.
Step 10: Multiply these values together: 15 * 16 * 9.
Step 11: Calculate the final result: 15 * 16 = 240, and then 240 * 9 = 2160.
Step 12: Conclude that the coefficient of x^4 in the expansion of (2x + 3)^6 is 2160.

Binomial Expansion – The question tests the understanding of the binomial theorem, which is used to expand expressions of the form (a + b)^n.
Combination Formula – The use of the combination formula C(n, k) to determine the number of ways to choose k successes in n trials is essential for finding the coefficient.
Power of a Product – Understanding how to calculate powers of coefficients (like (2)^4 and (3)^(6-4)) is crucial for determining the final coefficient.