What is the value of the coefficient of x^4 in the expansion of (3x - 2)^6?

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Question: What is the value of the coefficient of x^4 in the expansion of (3x - 2)^6?

Options:

Correct Answer: -540

Solution:

The coefficient of x^4 in (3x - 2)^6 is given by 6C4 * (3^4) * (-2)^2 = 15 * 81 * 4 = 4860.

What is the value of the coefficient of x^4 in the expansion of (3x - 2)^6?

What is the value of the coefficient of x^4 in the expansion of (3x - 2)^6?

Step 1: Identify the expression we need to expand, which is (3x - 2)^6.
Step 2: Recognize that we want the coefficient of x^4 in this expansion.
Step 3: Use the Binomial Theorem, which states that (a + b)^n = sum from k=0 to n of (nCk * a^(n-k) * b^k).
Step 4: In our case, a = 3x, b = -2, and n = 6.
Step 5: We need to find the term where the power of x is 4. This happens when (3x) is raised to the power of 4.
Step 6: To find this term, we set k = 2 because (n-k) = 4 means k = 6 - 4 = 2.
Step 7: Calculate the binomial coefficient 6C2, which is the number of ways to choose 2 from 6.
Step 8: 6C2 = 6! / (2! * (6-2)!) = 6! / (2! * 4!) = (6*5)/(2*1) = 15.
Step 9: Calculate (3x)^4, which is (3^4) * (x^4) = 81 * x^4.
Step 10: Calculate (-2)^2, which is 4.
Step 11: Now combine these results: Coefficient = 6C2 * (3^4) * (-2)^2 = 15 * 81 * 4.
Step 12: Calculate 15 * 81 = 1215.
Step 13: Now multiply 1215 by 4 to get 4860.
Step 14: Therefore, the coefficient of x^4 in the expansion of (3x - 2)^6 is 4860.

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