Find the coefficient of x^5 in the expansion of (2x - 3)^7. (2023)

Find the coefficient of x^5 in the expansion of (2x - 3)^7. (2023)

Step 1: Identify the expression we need to expand, which is (2x - 3)^7.
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 = 7.
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 = 2 (since 7 - 5 = 2).
Step 6: Calculate the binomial coefficient C(7, 2), which is the number of ways to choose 2 from 7.
Step 7: C(7, 2) = 7! / (2!(7-2)!) = 7! / (2! * 5!) = (7 * 6) / (2 * 1) = 21.
Step 8: Now calculate (2x)^5 = (2^5)(x^5) = 32x^5.
Step 9: Next, calculate (-3)^2 = 9.
Step 10: Combine these results to find the coefficient: Coefficient = C(7, 2) * (2^5) * (-3)^2 = 21 * 32 * 9.
Step 11: Finally, calculate 21 * 32 = 672, and then 672 * 9 = 6048.

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 (n choose k) is essential for finding the coefficients in the expansion.
Negative Exponents – Recognizing how to handle negative terms in the expansion, particularly when raised to a power.