What is the 4th term in the expansion of (3x + 2)^6?

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Question: What is the 4th term in the expansion of (3x + 2)^6?

Options:

Correct Answer: 540x^4

Solution:

The 4th term is given by C(6,3) * (3x)^3 * (2)^3 = 20 * 27x^3 * 8 = 4320x^3.

What is the 4th term in the expansion of (3x + 2)^6?

What is the 4th term in the expansion of (3x + 2)^6?

Step 1: Identify the expression to expand, which is (3x + 2)^6.
Step 2: Understand that we need to find the 4th term in the expansion.
Step 3: Use the binomial theorem, which states that the nth term in the expansion of (a + b)^n is given by C(n, k) * a^(n-k) * b^k, where C(n, k) is the binomial coefficient.
Step 4: For the 4th term, we need k = 3 (since we start counting from k = 0).
Step 5: Calculate n - k, which is 6 - 3 = 3.
Step 6: Identify a and b from the expression: a = 3x and b = 2.
Step 7: Calculate the binomial coefficient C(6, 3). This is equal to 6! / (3! * (6-3)!) = 20.
Step 8: Calculate (3x)^(3) = 27x^3.
Step 9: Calculate (2)^(3) = 8.
Step 10: Multiply the results: 20 * 27x^3 * 8.
Step 11: Calculate 20 * 27 = 540.
Step 12: Finally, calculate 540 * 8 = 4320.
Step 13: Therefore, the 4th term in the expansion is 4320x^3.

Binomial Expansion – The question tests the understanding of the binomial theorem, specifically how to find a specific term in the expansion of a binomial expression.
Combinatorial Coefficients – It requires knowledge of how to calculate binomial coefficients, which represent the number of ways to choose terms from the expansion.
Power of Terms – The question involves raising terms to specific powers and multiplying them correctly.