In the expansion of (2x - 3y)^5, what is the sign of the term containing x^3y^2?

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Q: In the expansion of (2x - 3y)^5, what is the sign of the term containing x^3y^2?

Solution: The term containing x^3y^2 will have a negative sign due to the -3y factor raised to an even power.

Steps: 12

Step 1: Identify the expression we are expanding, which is (2x - 3y)^5.
Step 2: Recognize that we are looking for the term that contains x^3y^2.
Step 3: Use the Binomial Theorem to find the general term in the expansion, which is given by T(k) = C(n, k) * (a^(n-k)) * (b^k), where n is the power, a is the first term, b is the second term, and C(n, k) is the binomial coefficient.
Step 4: In our case, n = 5, a = 2x, and b = -3y.
Step 5: We need to find k such that the term has x^3 and y^2. This means we want (2x)^(5-k) to give us x^3 and (-3y)^k to give us y^2.
Step 6: Set up the equations: 5 - k = 3 (for x) and k = 2 (for y).
Step 7: Solve these equations: From 5 - k = 3, we find k = 2.
Step 8: Now, substitute k = 2 into the term: T(2) = C(5, 2) * (2x)^(5-2) * (-3y)^2.
Step 9: Calculate C(5, 2) = 10, (2x)^3 = 8x^3, and (-3y)^2 = 9y^2.
Step 10: Combine these results: T(2) = 10 * 8x^3 * 9y^2 = 720x^3y^2.
Step 11: Notice that the term includes (-3y) raised to an even power (2), which means it will be positive.
Step 12: Therefore, the sign of the term containing x^3y^2 is positive.