Determine the coefficient of x^2 in the expansion of (3x - 4)^6.

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Question: Determine the coefficient of x^2 in the expansion of (3x - 4)^6.

Options:

Correct Answer: 720

Solution:

The coefficient of x^2 is C(6,2) * (3)^2 * (-4)^4 = 15 * 9 * 256 = 34560.

Determine the coefficient of x^2 in the expansion of (3x - 4)^6.

Determine the coefficient of x^2 in the expansion of (3x - 4)^6.

Correct Answer: 34560

Step 1: Identify the expression we need to expand, which is (3x - 4)^6.
Step 2: Recognize that we want 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 = 3x, b = -4, and n = 6.
Step 5: We need to find the term where the power of x is 2. This happens when (3x) is raised to the power of 2.
Step 6: Set k = 4 because we need (3x)^(2) and (-4)^(4) to complete the expansion to the 6th power (2 + 4 = 6).
Step 7: Calculate C(6, 2), which is the number of ways to choose 2 items from 6. This is equal to 15.
Step 8: Calculate (3)^2, which is 9.
Step 9: Calculate (-4)^4, which is 256.
Step 10: Multiply these values together: 15 (from C(6, 2)) * 9 (from (3)^2) * 256 (from (-4)^4).
Step 11: Perform the multiplication: 15 * 9 = 135, then 135 * 256 = 34560.
Step 12: The final result is that the coefficient of x^2 in the expansion of (3x - 4)^6 is 34560.

Binomial Expansion – The process of expanding expressions of the form (a + b)^n using the binomial theorem.
Coefficient Extraction – Identifying the specific coefficient of a term in a polynomial expansion.
Combinatorial Coefficients – Using combinations to determine the number of ways to choose terms in the expansion.