What is the constant term in the expansion of (2x^2 - 3/x)^5? (2020)

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Q: What is the constant term in the expansion of (2x^2 - 3/x)^5? (2020)

Solution: The constant term occurs when 2k - 5 = 0, k = 2. The term is C(5,2)(-3)^3(2^2) = 10 * -27 * 4 = -1080.

Steps: 13

Step 1: Identify the expression to expand, which is (2x^2 - 3/x)^5.
Step 2: Use the Binomial Theorem to expand the expression. The general term in the expansion is given by C(n, k) * (first term)^(n-k) * (second term)^k, where n is the power and k is the term number.
Step 3: In our case, n = 5, the first term is 2x^2, and the second term is -3/x.
Step 4: The general term can be written as C(5, k) * (2x^2)^(5-k) * (-3/x)^k.
Step 5: Simplify the general term: C(5, k) * (2^(5-k)) * (x^(2(5-k))) * (-3)^k * (x^(-k)).
Step 6: Combine the x terms: x^(2(5-k) - k) = x^(10 - 2k - k) = x^(10 - 3k).
Step 7: To find the constant term, set the exponent of x to 0: 10 - 3k = 0.
Step 8: Solve for k: 10 = 3k, so k = 10/3, which is not an integer. We need to find the integer k that gives a constant term.
Step 9: Check integer values for k. The constant term occurs when 2k - 5 = 0, which gives k = 2.
Step 10: Substitute k = 2 into the general term: C(5, 2) * (2x^2)^(3) * (-3/x)^(2).
Step 11: Calculate C(5, 2) = 10, (2^2) = 4, and (-3)^3 = -27.
Step 12: Combine these values: 10 * -27 * 4 = -1080.
Step 13: The constant term in the expansion is -1080.