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

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

Options:

Correct Answer: -36

Solution:

The coefficient of x^2 is given by 4C2 * (3^2) * (-2)^2 = 6 * 9 * 4 = 216.

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

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

Step 1: Identify the expression to expand, which is (3x - 2)^4.
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 = 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 = 4.
Step 5: We need to find the term where the power of x is 2. This happens when k = 2 because (3x)^(2) will give us x^2.
Step 6: Calculate nCk, which is 4C2. This is the number of ways to choose 2 items from 4, which equals 6.
Step 7: Calculate (3x)^(2) = (3^2)(x^2) = 9x^2.
Step 8: Calculate (-2)^(2) = 4.
Step 9: Multiply the results from steps 6, 7, and 8: 6 * 9 * 4.
Step 10: Perform the multiplication: 6 * 9 = 54, then 54 * 4 = 216.
Step 11: The coefficient of x^2 in the expansion is 216.

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