In the expansion of (2x + 3)^4, what is the coefficient of x^0?

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Question: In the expansion of (2x + 3)^4, what is the coefficient of x^0?

Options:

Correct Answer: 81

Solution:

The coefficient of x^0 is given by 4C4 * (2x)^0 * (3)^4 = 1 * 81 = 81.

In the expansion of (2x + 3)^4, what is the coefficient of x^0?

In the expansion of (2x + 3)^4, what is the coefficient of x^0?

Step 1: Understand that x^0 means we are looking for the constant term in the expansion of (2x + 3)^4.
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: Identify a = 2x, b = 3, and n = 4 in our case.
Step 4: To find the coefficient of x^0, we need to set (2x) to the power of 0. This means we will use k = 4 (the term with b^4).
Step 5: Calculate the binomial coefficient for k = 4: 4C4 = 1.
Step 6: Calculate (2x)^0, which equals 1.
Step 7: Calculate 3^4, which equals 81.
Step 8: Multiply the results from steps 5, 6, and 7: 1 * 1 * 81 = 81.
Step 9: Conclude that the coefficient of x^0 in the expansion of (2x + 3)^4 is 81.

Binomial Expansion – The question tests understanding of the binomial theorem, specifically how to find coefficients in the expansion of a binomial expression.
Coefficient Extraction – It assesses the ability to identify and calculate the coefficient of a specific term (x^0) in the expansion.