What is the term containing x^5 in the expansion of (2x - 3)^8?

What is the term containing x^5 in the expansion of (2x - 3)^8?

Step 1: Identify the expression we are expanding, which is (2x - 3)^8.
Step 2: Recognize that we need to find the term that contains x^5.
Step 3: Use the binomial theorem, which states that (a + b)^n = sum(C(n, k) * a^(n-k) * b^k) for k = 0 to n.
Step 4: In our case, a = 2x, b = -3, and n = 8.
Step 5: To find the term with x^5, we need to set the exponent of x in (2x)^(n-k) equal to 5.
Step 6: This means we need (2x)^(n-k) = (2x)^(8-k) to have x^5, so we set 8-k = 5.
Step 7: Solve for k: 8 - k = 5 gives k = 3.
Step 8: Now we can find the term when k = 3 using the binomial formula: C(8, 3) * (2x)^(8-3) * (-3)^3.
Step 9: Calculate C(8, 3), which is 56.
Step 10: Calculate (2x)^5, which is 32x^5.
Step 11: Calculate (-3)^3, which is -27.
Step 12: Combine these values: 56 * 32 * (-27).
Step 13: Calculate the final result: 56 * 32 = 1792, and then 1792 * (-27) = -48384.

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