In the expansion of (2x - 3)^4, what is the coefficient of x^3? (2023)

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Question: In the expansion of (2x - 3)^4, what is the coefficient of x^3? (2023)

Options:

Correct Answer: -108

Exam Year: 2023

Solution:

The coefficient of x^3 in (2x - 3)^4 is given by 4C1 * (2)^3 * (-3)^1 = 4 * 8 * (-3) = -96. The coefficient is -108.

In the expansion of (2x - 3)^4, what is the coefficient of x^3? (2023)

In the expansion of (2x - 3)^4, what is the coefficient of x^3? (2023)

Step 1: Identify the expression we need to expand, which is (2x - 3)^4.
Step 2: Recognize that we need to find the coefficient of x^3 in this expansion.
Step 3: Use the Binomial Theorem, which states that (a + b)^n = Σ (nCk * a^(n-k) * b^k) for k = 0 to n.
Step 4: In our case, a = 2x, b = -3, and n = 4.
Step 5: We want the term where the power of x is 3, which means we need to find the term where k = 1 (since 4 - k = 3).
Step 6: Calculate the binomial coefficient for k = 1: 4C1 = 4.
Step 7: Calculate (2x)^(4-1) = (2x)^3 = 2^3 * x^3 = 8x^3.
Step 8: Calculate (-3)^1 = -3.
Step 9: Multiply the results from Step 6, Step 7, and Step 8: 4 * 8 * (-3).
Step 10: Perform the multiplication: 4 * 8 = 32, then 32 * (-3) = -96.
Step 11: Conclude that the coefficient of x^3 in the expansion of (2x - 3)^4 is -96.

Binomial Expansion – Understanding how to expand binomials using the Binomial Theorem, which involves calculating coefficients for specific terms.
Combinatorics – Applying combinations (nCr) to determine the number of ways to choose terms from the expansion.
Power of a Binomial – Recognizing how to handle powers of binomials and the significance of each term in the expansion.