What is the term independent of x in the expansion of (3x - 2)^6?

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Question: What is the term independent of x in the expansion of (3x - 2)^6?

Options:

Correct Answer: 40

Solution:

The term independent of x occurs when k = 3. Using the binomial theorem, the term is C(6,3) * (3x)^3 * (-2)^(6-3) = 20 * 27 * -8 = -4320.

What is the term independent of x in the expansion of (3x - 2)^6?

What is the term independent of x in the expansion of (3x - 2)^6?

Step 1: Identify the expression to expand, which is (3x - 2)^6.
Step 2: 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 3: In our case, a = 3x, b = -2, and n = 6.
Step 4: We need to find the term that does not contain x, which means we want the power of x to be zero.
Step 5: The term (3x)^(n-k) will be x raised to the power of (n-k). We set (n-k) = 0 to find k.
Step 6: Since n = 6, we have 6 - k = 0, which means k = 6.
Step 7: The term independent of x occurs when k = 3 because we want (3x)^3 to balance with (-2)^(6-3).
Step 8: Calculate the coefficient using C(6, 3), which is the number of combinations of 6 items taken 3 at a time.
Step 9: C(6, 3) = 20.
Step 10: Now calculate the term: C(6, 3) * (3x)^3 * (-2)^(6-3).
Step 11: Substitute the values: 20 * (3^3) * (-2)^3.
Step 12: Calculate (3^3) = 27 and (-2)^3 = -8.
Step 13: Now multiply: 20 * 27 * -8.
Step 14: First, calculate 20 * 27 = 540.
Step 15: Then multiply 540 * -8 = -4320.
Step 16: The term independent of x in the expansion is -4320.

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