Find the coefficient of x^0 in the expansion of (2x - 3)^3.

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Question: Find the coefficient of x^0 in the expansion of (2x - 3)^3.

Options:

Correct Answer: -27

Solution:

The coefficient of x^0 is (-3)^3 = -27.

Find the coefficient of x^0 in the expansion of (2x - 3)^3.

Find the coefficient of x^0 in the expansion of (2x - 3)^3.

Step 1: Understand that x^0 means we are looking for the constant term in the expansion.
Step 2: Use the binomial theorem to expand (2x - 3)^3.
Step 3: The binomial theorem states that (a + b)^n = sum of (n choose k) * a^(n-k) * b^k for k from 0 to n.
Step 4: In our case, a = 2x, b = -3, and n = 3.
Step 5: We need to find the term where x has an exponent of 0, which happens when we take b^3.
Step 6: The term we need is (-3)^3, since we take all of b and none of a.
Step 7: Calculate (-3)^3 = -27.
Step 8: Therefore, the coefficient of x^0 in the expansion is -27.

Binomial Expansion – Understanding how to expand expressions of the form (a + b)^n using the binomial theorem.
Finding Coefficients – Identifying specific coefficients in the expansion, particularly the constant term (x^0).