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

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

Options:

Correct Answer: 216

Solution:

The term containing x^2 is given by C(4,2) * (3x)^2 * 4^2 = 6 * 9 * 16 = 864.

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

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

Step 1: Identify the expression to expand, which is (3x + 4)^4.
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 = 4, and n = 4.
Step 4: We want the term that contains x^2. This occurs when we choose (3x) two times and 4 two times.
Step 5: Calculate the binomial coefficient C(4, 2), which is the number of ways to choose 2 items from 4. C(4, 2) = 4! / (2! * (4-2)!) = 6.
Step 6: Calculate (3x)^2, which is (3^2)(x^2) = 9x^2.
Step 7: Calculate 4^2, which is 16.
Step 8: Combine these results: The term containing x^2 is C(4, 2) * (3x)^2 * 4^2 = 6 * 9 * 16.
Step 9: Perform the multiplication: 6 * 9 = 54, and then 54 * 16 = 864.
Step 10: Therefore, the term containing x^2 in the expansion is 864.

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