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

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

Options:

Correct Answer: 216

Solution:

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

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

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

Correct Answer: 864

Step 1: Identify the expression to expand, which is (3x - 4)^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 = 3x, b = -4, and n = 4.
Step 5: We want the term where the power of x is 2, which means we need (3x)^(2) and (-4)^(4-2).
Step 6: Set k = 2, since we want the x^2 term. Calculate C(4, 2), which is the number of ways to choose 2 from 4.
Step 7: C(4, 2) = 4! / (2! * (4-2)!) = 6.
Step 8: Calculate (3)^2, which is 9.
Step 9: Calculate (-4)^2, which is 16.
Step 10: Multiply these values together: 6 (from C(4, 2)) * 9 (from (3)^2) * 16 (from (-4)^2).
Step 11: Perform the multiplication: 6 * 9 = 54, then 54 * 16 = 864.
Step 12: The coefficient of x^2 in the expansion of (3x - 4)^4 is 864.

Binomial Expansion – The process of expanding expressions that are raised to a power, using the Binomial Theorem.
Coefficients in Polynomial Expansion – Understanding how to find specific coefficients in the expansion of polynomials.
Combinatorial Coefficients – Using combinations to determine the number of ways to choose terms from the expansion.