In the expansion of (3x - 4)^4, what is the coefficient of x^2? (2023)

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Question: In the expansion of (3x - 4)^4, what is the coefficient of x^2? (2023)

Options:

Correct Answer: -216

Solution:

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

In the expansion of (3x - 4)^4, what is the coefficient of x^2? (2023)

In the expansion of (3x - 4)^4, what is the coefficient of x^2? (2023)

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 the 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 to find the term where (3x) is raised to the power of 2.
Step 6: This corresponds to k = 2 in the binomial expansion, since we want (3x)^(4-2) and (-4)^2.
Step 7: Calculate C(4, 2), which is the number of ways to choose 2 from 4. 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: Coefficient = C(4, 2) * (3)^2 * (-4)^2 = 6 * 9 * 16.
Step 11: Perform the multiplication: 6 * 9 = 54, and then 54 * 16 = 864.
Step 12: Since we have (-4)^2, the coefficient is positive, so the final coefficient of x^2 is 864.

Binomial Expansion – The question tests the understanding of the binomial theorem, specifically how to find coefficients in the expansion of a binomial expression.
Combinatorics – It involves calculating combinations, as the coefficient of a term in the expansion is determined by the binomial coefficient.
Exponent Rules – Understanding how to apply exponent rules when expanding the terms in the binomial expression.