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

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

Correct Answer: 34560

Step 1: Identify the expression to expand, which is (3x - 4)^6.
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 = 6.
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 = 4 because (n - k) = 2 (since 6 - 4 = 2).
Step 6: Calculate C(6, 4), which is the number of ways to choose 4 from 6. C(6, 4) = 6! / (4! * (6-4)!) = 15.
Step 7: Calculate (3)^2, which is 9.
Step 8: Calculate (-4)^4, which is 256.
Step 9: Multiply these values together: Coefficient = C(6, 4) * (3)^2 * (-4)^4 = 15 * 9 * 256.
Step 10: Perform the multiplication: 15 * 9 = 135, then 135 * 256 = 34560.
Step 11: The coefficient of x^2 in the expansion of (3x - 4)^6 is 34560.

Binomial Theorem – The Binomial Theorem is used to expand expressions of the form (a + b)^n, where the coefficients can be calculated using combinations.
Combinations – Understanding how to calculate combinations (C(n, k)) is essential for determining the coefficients in the binomial expansion.
Negative Exponents – Recognizing how to handle negative terms in the expansion, particularly when raised to an even power.