In the expansion of (2x - 3)^6, what is the term containing x^4?

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Question: In the expansion of (2x - 3)^6, what is the term containing x^4?

Options:

Correct Answer: -540x^4

Solution:

The term containing x^4 is given by 6C4 * (2^4) * (-3)^2 = 15 * 16 * 9 = -2160.

In the expansion of (2x - 3)^6, what is the term containing x^4?

In the expansion of (2x - 3)^6, what is the term containing x^4?

Step 1: Identify the expression we are expanding, which is (2x - 3)^6.
Step 2: Use the Binomial Theorem, which states that (a + b)^n = Σ (nCk * a^(n-k) * b^k) for k = 0 to n.
Step 3: In our case, a = 2x, b = -3, and n = 6.
Step 4: We want the term that contains x^4. This means we need to find the term where (2x) has the power of 4.
Step 5: If (2x) has the power of 4, then (-3) must have the power of (6 - 4) = 2.
Step 6: The term we are looking for is given by the formula: nCk * (2x)^(n-k) * (-3)^k, where k = 2.
Step 7: Calculate nCk, which is 6C2. This is equal to 15.
Step 8: Calculate (2x)^4, which is (2^4) * (x^4) = 16 * x^4.
Step 9: Calculate (-3)^2, which is 9.
Step 10: Combine these results: 15 * 16 * 9 = 2160.
Step 11: Since we have (-3)^2, the term is positive, so the final term containing x^4 is 2160.

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