Determine the coefficient of x^4 in the expansion of (2x - 3)^6.

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Question: Determine the coefficient of x^4 in the expansion of (2x - 3)^6.

Options:

Correct Answer: 720

Solution:

The coefficient of x^4 is given by 6C4 * (2)^4 * (-3)^2 = 15 * 16 * 9 = 2160.

Determine the coefficient of x^4 in the expansion of (2x - 3)^6.

Determine the coefficient of x^4 in the expansion of (2x - 3)^6.

Step 1: Identify the expression to expand, which is (2x - 3)^6.
Step 2: Use the Binomial Theorem, which states that (a + b)^n = sum of (nCk * a^(n-k) * b^k) for k from 0 to n.
Step 3: In our case, a = 2x, b = -3, and n = 6.
Step 4: We want the coefficient of x^4, which means we need to find the term where (2x) is raised to the power of 4.
Step 5: This occurs when k = 2 because (n-k) = 4, so k = n - 4 = 6 - 4 = 2.
Step 6: Calculate the binomial coefficient 6C2, which is the number of ways to choose 2 from 6.
Step 7: 6C2 = 6! / (2! * (6-2)!) = 6! / (2! * 4!) = (6*5)/(2*1) = 15.
Step 8: Calculate (2)^4, which is 16.
Step 9: Calculate (-3)^2, which is 9.
Step 10: Multiply the results: Coefficient = 6C2 * (2)^4 * (-3)^2 = 15 * 16 * 9.
Step 11: Calculate 15 * 16 = 240.
Step 12: Then calculate 240 * 9 = 2160.
Step 13: The coefficient of x^4 in the expansion is 2160.

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 (nCr) is essential for determining the coefficients in the expansion.
Negative Exponents – Recognizing how to handle negative numbers in the expansion, particularly when raised to an even power.