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

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

Options:

Correct Answer: 1080

Solution:

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

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

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

Step 1: Identify the expression to expand, which is (3x - 4)^6.
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 = 6.
Step 5: We want the term where the power of x is 2, which means we need to find the term where (3x) is raised to the power of 2.
Step 6: This corresponds to k = 6 - 2 = 4, since we want the x^2 term.
Step 7: Calculate C(6, 2), which is the number of ways to choose 2 from 6. C(6, 2) = 6! / (2!(6-2)!) = 15.
Step 8: Calculate (3)^2, which is 9.
Step 9: Calculate (-4)^(6-2), which is (-4)^4. This equals 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, and then 135 * 256 = 34560.
Step 12: The coefficient of x^2 in the expansion of (3x - 4)^6 is 34560.

Binomial Expansion – The question tests the understanding of the binomial theorem, specifically how to find specific coefficients in the expansion of a binomial expression.
Combinatorics – The use of combinations (C(n, k)) to determine the number of ways to choose terms from the expansion.
Exponent Rules – Applying exponent rules correctly when calculating powers of the terms in the binomial.