Find the coefficient of x^5 in the expansion of (2x - 3)^8.

Find the coefficient of x^5 in the expansion of (2x - 3)^8.

Step 1: Identify the expression we need to expand, which is (2x - 3)^8.
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 = 2x, b = -3, and n = 8.
Step 4: We want the term that contains x^5. This occurs when (2x) is raised to the power of 5.
Step 5: To find the corresponding k, we set n - k = 5, which gives us k = 8 - 5 = 3.
Step 6: Calculate the binomial coefficient C(8, 3), which is the number of ways to choose 3 from 8.
Step 7: C(8, 3) = 8! / (3! * (8-3)!) = 56.
Step 8: Now calculate (2x)^5 = (2^5)(x^5) = 32x^5.
Step 9: Next, calculate (-3)^3 = -27.
Step 10: Combine these results to find the coefficient of x^5: C(8, 3) * (2^5) * (-3)^3 = 56 * 32 * (-27).
Step 11: Calculate 56 * 32 = 1792.
Step 12: Now multiply 1792 by -27 to get -6720.
Step 13: Therefore, the coefficient of x^5 in the expansion of (2x - 3)^8 is -6720.

Binomial Theorem – The Binomial Theorem is used to expand expressions of the form (a + b)^n, where the coefficients can be determined using combinations.
Combinations – Understanding how to calculate combinations (C(n, k)) is essential for finding the coefficients in the binomial expansion.
Negative Exponents – Recognizing how to handle negative terms in the expansion, particularly when raised to a power.