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

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

Options:

Correct Answer: 432

Solution:

The coefficient of x^2 is C(4,2) * (3)^2 * (-4)^2 = 6 * 9 * 16 = 864.

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

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

Step 1: Identify the expression to expand, which is (3x - 4)^4.
Step 2: 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 3: In our case, a = 3x, b = -4, and n = 4.
Step 4: We want the coefficient of x^2, which means we need the term where (3x) is raised to the power of 2.
Step 5: This occurs when k = 2, because (3x)^(2) corresponds to x^2.
Step 6: Calculate C(4, 2), which is the number of ways to choose 2 from 4. C(4, 2) = 4! / (2! * (4-2)!) = 6.
Step 7: Calculate (3)^2, which is 9.
Step 8: Calculate (-4)^2, which is 16.
Step 9: Multiply these values together: Coefficient = C(4, 2) * (3)^2 * (-4)^2 = 6 * 9 * 16.
Step 10: Perform the multiplication: 6 * 9 = 54, then 54 * 16 = 864.
Step 11: The coefficient of x^2 in the expansion of (3x - 4)^4 is 864.

Binomial Expansion – The question tests the understanding of the binomial theorem, specifically how to find the coefficient of a specific term in the expansion of a binomial expression.
Combinatorial Coefficients – It requires knowledge of how to calculate combinations, denoted as C(n, k), which represents the number of ways to choose k elements from a set of n elements.
Power of a Binomial – The question involves raising a binomial to a power and understanding how to apply the theorem to extract specific coefficients.