Which of the following expressions represents the coefficient of x^3 in the expa

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Question: Which of the following expressions represents the coefficient of x^3 in the expansion of (2x + 3)^5?

Options:

Correct Answer: 90

Solution:

Using the Binomial Theorem, the coefficient of x^3 is given by C(5,3) * (2^3) * (3^2) = 10 * 8 * 9 = 720.

Which of the following expressions represents the coefficient of x^3 in the expansion of (2x + 3)^5?

Which of the following expressions represents the coefficient of x^3 in the expansion of (2x + 3)^5?

Step 1: Identify the expression we are expanding, which is (2x + 3)^5.
Step 2: Recognize that we need to find the coefficient of x^3 in this 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 = 2x, b = 3, and n = 5.
Step 5: We want the term where x has the power of 3, which means we need to find the term where (2x) is raised to the power of 3.
Step 6: This means we need to find the term where k = 2 (because n - k = 3, so k = 5 - 3 = 2).
Step 7: Calculate C(5, 2), which is the number of ways to choose 2 from 5. C(5, 2) = 5! / (2!(5-2)!) = 10.
Step 8: Calculate (2^3), which is 2 raised to the power of 3. 2^3 = 8.
Step 9: Calculate (3^2), which is 3 raised to the power of 2. 3^2 = 9.
Step 10: Multiply these values together: Coefficient = C(5, 2) * (2^3) * (3^2) = 10 * 8 * 9.
Step 11: Perform the multiplication: 10 * 8 = 80, and then 80 * 9 = 720.
Step 12: Conclude that the coefficient of x^3 in the expansion of (2x + 3)^5 is 720.

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