What is the 3rd term in the expansion of (x + 3)^5? (2023)

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Q: What is the 3rd term in the expansion of (x + 3)^5? (2023)

Solution: The 3rd term is C(5,2) * (3)^2 * (x)^3 = 10 * 9 * x^3 = 90.

Steps: 11

Step 1: Identify the expression to expand, which is (x + 3)^5.
Step 2: Understand that we want the 3rd term in the expansion.
Step 3: Use the binomial theorem, which states that the nth term in the expansion of (a + b)^n is given by C(n, k) * a^(n-k) * b^k, where C(n, k) is the binomial coefficient.
Step 4: For the 3rd term, we need to find k = 2 (since we start counting from k = 0).
Step 5: Calculate C(5, 2), which is the number of ways to choose 2 items from 5. This is calculated as 5! / (2!(5-2)!) = 10.
Step 6: Identify a and b in our expression: a = x and b = 3.
Step 7: Calculate a^(n-k) = x^(5-2) = x^3.
Step 8: Calculate b^k = 3^2 = 9.
Step 9: Combine these results: The 3rd term = C(5, 2) * (3^2) * (x^3) = 10 * 9 * x^3.
Step 10: Multiply the coefficients: 10 * 9 = 90.
Step 11: Write the final answer: The 3rd term is 90 * x^3.