What is the 3rd term in the expansion of (2x - 3)^5? (2022)

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Question: What is the 3rd term in the expansion of (2x - 3)^5? (2022)

Options:

Correct Answer: -90x^3

Exam Year: 2022

Solution:

The 3rd term is C(5,2) * (2x)^3 * (-3)^2 = 10 * 8x^3 * 9 = -720x^3.

What is the 3rd term in the expansion of (2x - 3)^5? (2022)

What is the 3rd term in the expansion of (2x - 3)^5? (2022)

Step 1: Identify the expression to expand, which is (2x - 3)^5.
Step 2: Recognize that we need to find 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 T(k+1) = C(n, k) * a^(n-k) * b^k.
Step 4: In our case, a = 2x, b = -3, and n = 5.
Step 5: For the 3rd term, k = 2 (since we start counting from 0).
Step 6: Calculate C(5, 2), which is the number of combinations of 5 items taken 2 at a time. C(5, 2) = 5! / (2!(5-2)!) = 10.
Step 7: Calculate (2x)^(5-2) = (2x)^3 = 8x^3.
Step 8: Calculate (-3)^2 = 9.
Step 9: Multiply the results: 10 * 8x^3 * 9 = 720x^3.
Step 10: Since the term involves (-3), the final result is -720x^3.

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