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

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

Options:

Correct Answer: 15

Solution:

The coefficient of x is given by C(5, 1) * (2)^4 * (3) = 5 * 16 * 3 = 240.

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

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

Step 1: Identify the expression to expand, which is (2 + 3x)^5.
Step 2: Recognize that we need to find the coefficient of x 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 = 2, b = 3x, and n = 5.
Step 5: We want the term where x is to the power of 1, which means we need k = 1.
Step 6: Calculate C(5, 1), which is the number of ways to choose 1 item from 5. This equals 5.
Step 7: Calculate (2)^(5-1) = (2)^4 = 16.
Step 8: Calculate (3)^(1) = 3.
Step 9: Multiply these values together: 5 * 16 * 3.
Step 10: Perform the multiplication: 5 * 16 = 80, then 80 * 3 = 240.
Step 11: Conclude that the coefficient of x in the expansion is 240.

Binomial Expansion – The question tests understanding of the binomial theorem, specifically how to find coefficients in the expansion of a binomial expression.
Combinatorics – It involves calculating combinations, specifically C(5, 1), which represents the number of ways to choose 1 term from 5.
Polynomial Coefficients – The focus is on identifying the coefficient of a specific term (x) in the polynomial expansion.