What is the 5th term in the expansion of (3x - 2)^6?

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Question: What is the 5th term in the expansion of (3x - 2)^6?

Options:

Correct Answer: -540x^5

Solution:

The 5th term is given by C(6, 4) * (3x)^4 * (-2)^2 = 15 * 81x^4 * 4 = 4860x^4.

What is the 5th term in the expansion of (3x - 2)^6?

What is the 5th term in the expansion of (3x - 2)^6?

Step 1: Identify the expression to expand, which is (3x - 2)^6.
Step 2: Determine the term number we want, which is the 5th term.
Step 3: Use the binomial expansion formula to find the 5th term. The formula is C(n, k) * a^(n-k) * b^k, where n is the exponent, k is the term number minus 1, a is the first term, and b is the second term.
Step 4: For the 5th term, n = 6 and k = 4 (since we start counting from 0).
Step 5: Calculate C(6, 4), which is the number of combinations of 6 items taken 4 at a time. This equals 15.
Step 6: Calculate (3x)^(6-4) = (3x)^2 = 9x^2.
Step 7: Calculate (-2)^4 = 16.
Step 8: Combine these results: 15 * 9x^2 * 16.
Step 9: Multiply the numbers: 15 * 9 = 135 and then 135 * 16 = 2160.
Step 10: The final result is 2160x^2.

Binomial Expansion – Understanding how to expand expressions of the form (a + b)^n using the binomial theorem.
Combination Formula – Using the combination formula C(n, k) to find the coefficients in the expansion.
Term Identification – Identifying the correct term in the expansion based on its position.