In the expansion of (2x - 3)^5, what is the coefficient of x^1?

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Question: In the expansion of (2x - 3)^5, what is the coefficient of x^1?

Options:

Correct Answer: -30

Solution:

The coefficient of x^1 is C(5,1) * (2)^1 * (-3)^4 = 5 * 2 * 81 = 810.

In the expansion of (2x - 3)^5, what is the coefficient of x^1?

In the expansion of (2x - 3)^5, what is the coefficient of x^1?

Step 1: Identify the expression to expand, which is (2x - 3)^5.
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 [C(n, k) * a^(n-k) * b^k] for k = 0 to n.
Step 4: In our case, a = 2x, b = -3, and n = 5.
Step 5: We want the term where the power of x is 1, which means we need to find the term where k = 4 (since 5 - k = 1).
Step 6: Calculate C(5, 4), which is the number of ways to choose 4 from 5. This equals 5.
Step 7: Calculate (2)^1, which is 2.
Step 8: Calculate (-3)^4, which is 81 (since -3 * -3 * -3 * -3 = 81).
Step 9: Multiply these values together: 5 (from C(5, 4)) * 2 (from (2)^1) * 81 (from (-3)^4).
Step 10: The final calculation is 5 * 2 * 81 = 810.

Binomial Expansion – The question tests the understanding of the binomial theorem, specifically how to find coefficients in the expansion of a binomial expression.
Combination Formula – It requires knowledge of combinations, specifically C(n, k), to determine the number of ways to choose k successes in n trials.
Power of a Binomial – The question involves calculating powers of terms in a binomial expression and applying the correct signs based on the terms.