What is the coefficient of x^4 in the expansion of (3x - 2)^5? (2021)

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Question: What is the coefficient of x^4 in the expansion of (3x - 2)^5? (2021)

Options:

Correct Answer: 150

Exam Year: 2021

Solution:

The coefficient of x^4 is given by 5C4 * (3)^4 * (-2)^1 = 5 * 81 * (-2) = -810.

What is the coefficient of x^4 in the expansion of (3x - 2)^5? (2021)

What is the coefficient of x^4 in the expansion of (3x - 2)^5? (2021)

Step 1: Identify the expression to expand, which is (3x - 2)^5.
Step 2: Use the Binomial Theorem, which states that (a + b)^n = Σ (nCk * a^(n-k) * b^k) for k = 0 to n.
Step 3: In our case, a = 3x, b = -2, and n = 5.
Step 4: We want the coefficient of x^4, which means we need the term where (3x) is raised to the power of 4.
Step 5: This occurs when k = 1 because (3x)^(5-k) = (3x)^4 when k = 1.
Step 6: Calculate nCk, which is 5C1. This equals 5.
Step 7: Calculate (3)^4, which is 81.
Step 8: Calculate (-2)^1, which is -2.
Step 9: Multiply these values together: 5 * 81 * (-2).
Step 10: The result is -810, which is the coefficient of x^4.

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