Find the coefficient of x^3 in the expansion of (2x - 3)^4. (2022)

Find the coefficient of x^3 in the expansion of (2x - 3)^4. (2022)

Step 1: Identify the expression to expand, which is (2x - 3)^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 = 2x, b = -3, and n = 4.
Step 4: We want the coefficient of x^3, which means we need the term where (2x) is raised to the power of 3.
Step 5: This occurs when k = 1 because n - k = 3 (4 - 1 = 3).
Step 6: Calculate C(4, 3), which is the number of ways to choose 3 items from 4. C(4, 3) = 4.
Step 7: Calculate (2)^3, which is 2 raised to the power of 3. (2)^3 = 8.
Step 8: Calculate (-3)^1, which is -3 raised to the power of 1. (-3)^1 = -3.
Step 9: Multiply the results from steps 6, 7, and 8: 4 * 8 * (-3).
Step 10: Calculate the final result: 4 * 8 = 32, and then 32 * (-3) = -96.

Binomial Expansion – The question tests the understanding of the binomial theorem, which allows for the expansion of expressions of the form (a + b)^n.
Combination Formula – The use of the combination formula C(n, k) to determine the number of ways to choose k successes in n trials is essential for finding specific coefficients.
Coefficient Extraction – The ability to extract the coefficient of a specific term (x^3) from the expanded form is a key skill in polynomial expansions.