Find the coefficient of x^3 in the expansion of (3x - 4)^5.

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Question: Find the coefficient of x^3 in the expansion of (3x - 4)^5.

Options:

Correct Answer: -720

Solution:

The coefficient of x^3 in (3x - 4)^5 is given by 5C3 * (3)^3 * (-4)^2 = 10 * 27 * 16 = -720.

Find the coefficient of x^3 in the expansion of (3x - 4)^5.

Find the coefficient of x^3 in the expansion of (3x - 4)^5.

Step 1: Identify the expression we need to expand, which is (3x - 4)^5.
Step 2: Use the Binomial Theorem, which states that (a + b)^n = sum of (nCk * a^(n-k) * b^k) for k from 0 to n.
Step 3: In our case, a = 3x, b = -4, and n = 5.
Step 4: We want the coefficient of x^3, which means we need to find the term where (3x) is raised to the power of 3.
Step 5: This occurs when k = 2 because (n-k) = 3 implies k = 5 - 3 = 2.
Step 6: Calculate the binomial coefficient for k = 2: 5C2 = 5! / (2!(5-2)!) = 10.
Step 7: Calculate (3x)^3: (3)^3 = 27 and x^3 remains as x^3.
Step 8: Calculate (-4)^2: (-4)^2 = 16.
Step 9: Multiply the results: Coefficient = 5C2 * (3)^3 * (-4)^2 = 10 * 27 * 16.
Step 10: Calculate 10 * 27 = 270, then 270 * 16 = 4320.
Step 11: Since we have (-4)^2, the final coefficient is positive, so the coefficient of x^3 is 4320.

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 binomial coefficients (nCr) to determine the number of ways to choose terms from the expansion.
Power of a Binomial – Understanding how to apply powers to each term in the binomial and how to calculate the resulting coefficients.