In the expansion of (3x - 2)^4, what is the coefficient of x^1?

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Question: In the expansion of (3x - 2)^4, what is the coefficient of x^1?

Options:

Correct Answer: -72

Solution:

The coefficient of x^1 in (3x - 2)^4 is given by 4C3 * (3x)^1 * (-2)^3 = 4 * 3 * (-8) = -96.

In the expansion of (3x - 2)^4, what is the coefficient of x^1?

In the expansion of (3x - 2)^4, what is the coefficient of x^1?

Step 1: Identify the expression to expand, which is (3x - 2)^4.
Step 2: Recognize that we need to find the coefficient of x^1 in the 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 = 3x, b = -2, and n = 4.
Step 5: We want the term where the power of x is 1, which means we need (3x)^1 and (-2)^(4-1) = (-2)^3.
Step 6: To find this term, we need to calculate the binomial coefficient for k = 3 (since we want x^1, which corresponds to k = n - 1).
Step 7: Calculate the binomial coefficient: 4C3 = 4! / (3! * (4-3)!) = 4.
Step 8: Now calculate the term: 4C3 * (3x)^1 * (-2)^3.
Step 9: Substitute the values: 4 * (3x) * (-8) (since (-2)^3 = -8).
Step 10: Simplify: 4 * 3 * (-8) = -96.
Step 11: The coefficient of x^1 in the expansion is -96.

Binomial Expansion – The question tests the understanding of the binomial theorem, specifically how to find coefficients in the expansion of a binomial expression.
Combinatorics – The use of binomial coefficients (nCr) to determine the number of ways to choose terms from the expansion.
Negative Exponents – Understanding how to handle negative terms in the expansion, particularly when calculating powers of negative numbers.