In the expansion of (3x - 2)^5, what is the coefficient of x^3?

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Question: In the expansion of (3x - 2)^5, what is the coefficient of x^3?

Options:

Correct Answer: -360

Solution:

The coefficient of x^3 in (3x - 2)^5 is given by 5C2 * (3)^3 * (-2)^2 = 10 * 27 * 4 = -1080.

In the expansion of (3x - 2)^5, what is the coefficient of x^3?

In the expansion of (3x - 2)^5, what is the coefficient of x^3?

Step 1: Identify the expression we are expanding, which is (3x - 2)^5.
Step 2: Recognize that we need to find the coefficient of x^3 in this expansion.
Step 3: 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 4: In our case, a = 3x, b = -2, and n = 5.
Step 5: We want the term where the power of x is 3, which means we need (3x)^(3) and (-2)^(5-3) = (-2)^(2).
Step 6: To find the correct term, we need to calculate k, where k = 5 - 3 = 2.
Step 7: Calculate the binomial coefficient, which is 5C2. This is equal to 5! / (2! * (5-2)!) = 10.
Step 8: Calculate (3)^3, which is 27.
Step 9: Calculate (-2)^2, which is 4.
Step 10: Multiply these values together: 5C2 * (3)^3 * (-2)^2 = 10 * 27 * 4.
Step 11: Perform the multiplication: 10 * 27 = 270, and then 270 * 4 = 1080.
Step 12: Since we are looking for the coefficient of x^3, the final answer is 1080.

Binomial Expansion – The question tests the understanding of the binomial theorem, specifically how to find 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.
Exponentiation – Understanding how to apply powers to coefficients and variables in the expansion.