What is the coefficient of x^0 in the expansion of (3x + 2)^5? (2022)

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Question: What is the coefficient of x^0 in the expansion of (3x + 2)^5? (2022)

Options:

Correct Answer: 64

Exam Year: 2022

Solution:

The coefficient of x^0 is C(5,5) * (3x)^0 * (2)^5 = 1 * 32 = 32.

What is the coefficient of x^0 in the expansion of (3x + 2)^5? (2022)

What is the coefficient of x^0 in the expansion of (3x + 2)^5? (2022)

Step 1: Understand that x^0 means we want the constant term in the expansion of (3x + 2)^5.
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: Identify a and b in our expression: a = 3x, b = 2, and n = 5.
Step 4: To find the coefficient of x^0, we need to set (3x)^(n-k) to (3x)^0. This means n-k must equal 0, so k must equal n, which is 5.
Step 5: Calculate C(5, 5), which is the number of ways to choose 5 items from 5. This equals 1.
Step 6: Calculate (3x)^0, which equals 1 because any number to the power of 0 is 1.
Step 7: Calculate (2)^5, which equals 32.
Step 8: Multiply the results from steps 5, 6, and 7: 1 * 1 * 32 = 32.
Step 9: Conclude that the coefficient of x^0 in the expansion of (3x + 2)^5 is 32.

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