Find the coefficient of x^3 in the expansion of (2x - 3)^6.

Find the coefficient of x^3 in the expansion of (2x - 3)^6.

Correct Answer: -4320

Step 1: Identify the expression to expand, which is (2x - 3)^6.
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 = 6.
Step 4: We want the term where the power of x is 3, which means we need to find the term where (2x) is raised to the power of 3.
Step 5: This corresponds to k = 3 in the Binomial Theorem, since we want (2x)^(6-3) and (-3)^3.
Step 6: Calculate C(6, 3), which is the number of ways to choose 3 items from 6. C(6, 3) = 20.
Step 7: Calculate (2)^3, which is 2 * 2 * 2 = 8.
Step 8: Calculate (-3)^3, which is -3 * -3 * -3 = -27.
Step 9: Multiply these values together: 20 * 8 * (-27).
Step 10: First, calculate 20 * 8 = 160.
Step 11: Then, calculate 160 * (-27) = -4320.
Step 12: The coefficient of x^3 in the expansion is -4320.

Binomial Theorem – The Binomial Theorem is used to expand expressions of the form (a + b)^n, where the coefficients can be calculated using combinations.
Combination Notation – Understanding how to calculate combinations, denoted as C(n, k), which represents the number of ways to choose k elements from a set of n elements.
Negative Exponents – Recognizing how to handle negative numbers in the expansion, particularly when raising a negative term to a power.