What is the value of the coefficient of x^5 in the expansion of (3x - 2)^8?

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Question: What is the value of the coefficient of x^5 in the expansion of (3x - 2)^8?

Options:

Correct Answer: -6720

Solution:

The coefficient is C(8,5) * (3^5) * (-2)^3 = 56 * 243 * (-8) = -6720.

What is the value of the coefficient of x^5 in the expansion of (3x - 2)^8?

What is the value of the coefficient of x^5 in the expansion of (3x - 2)^8?

Step 1: Identify the expression we need to expand, which is (3x - 2)^8.
Step 2: Recognize that we want the coefficient of x^5 in this expansion.
Step 3: 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 4: In our case, a = 3x, b = -2, and n = 8.
Step 5: We need to find the term where the power of x is 5. This happens when (3x) is raised to the power of 5.
Step 6: Set k = 3 because we need (3x)^(5) and (-2)^(3) to complete the expansion to 8.
Step 7: Calculate the binomial coefficient C(8, 5), which is the same as C(8, 3). This equals 56.
Step 8: Calculate (3^5), which is 243.
Step 9: Calculate (-2)^3, which is -8.
Step 10: Multiply these values together: 56 * 243 * (-8).
Step 11: First, calculate 56 * 243 = 13608.
Step 12: Then multiply 13608 by -8 to get -108864.
Step 13: Therefore, the coefficient of x^5 in the expansion of (3x - 2)^8 is -108864.

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