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

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

Options:

Correct Answer: -18

Solution:

The coefficient of x is C(3,1) * (2)^1 * (-3)^2 = 3 * 2 * 9 = -54.

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

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

Step 1: Identify the expression to expand, which is (2x - 3)^3.
Step 2: 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 3: In our case, a = 2x, b = -3, and n = 3.
Step 4: We want the coefficient of x, which corresponds to the term where the power of x is 1.
Step 5: To find this term, we need k = 2 (because we want (2x)^1, which means we need to use b^2).
Step 6: Calculate C(3, 1), which is the number of ways to choose 1 from 3, and it equals 3.
Step 7: Calculate (2)^1, which is 2.
Step 8: Calculate (-3)^2, which is 9.
Step 9: Multiply these values together: 3 (from C(3, 1)) * 2 (from (2)^1) * 9 (from (-3)^2).
Step 10: The result is 3 * 2 * 9 = 54.
Step 11: Since we are looking for the coefficient of x, we note that the term is negative, so the coefficient of x is -54.

Binomial Expansion – Understanding how to expand binomials using the Binomial Theorem, which involves calculating coefficients for specific terms.
Coefficient Calculation – Identifying and calculating the coefficient of a specific term in the expansion, particularly focusing on the linear term.