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Expand and simplify: the square of (3x - 5xy), written fully as a polynomial in x and y.

Difficulty: Easy

Correct Answer: 9x^2 - 30x^2*y + 25x^2*y^2

Explanation:


Introduction / Context:
This question tests the standard algebra identity for squaring a binomial: (a - b)^2 = a^2 - 2ab + b^2. Here, the terms are not simple numbers; they are algebraic expressions (3x and 5xy). The only real task is to apply the identity carefully and multiply coefficients and variables correctly.


Given Data / Assumptions:

  • Expression to square: (3x - 5xy)
  • Identity: (a - b)^2 = a^2 - 2ab + b^2
  • Let a = 3x and b = 5xy


Concept / Approach:
Compute a^2, compute 2ab, compute b^2, then combine with the correct signs: a^2 - 2ab + b^2.


Step-by-Step Solution:

Step 1: Let a = 3x and b = 5xy. Step 2: Compute a^2: (3x)^2 = 9x^2. Step 3: Compute 2ab: 2*(3x)*(5xy) = 2*15*x*x*y = 30x^2*y. Step 4: Compute b^2: (5xy)^2 = 25*x^2*y^2 = 25x^2*y^2. Step 5: Apply formula: (3x - 5xy)^2 = a^2 - 2ab + b^2 = 9x^2 - 30x^2*y + 25x^2*y^2.


Verification / Alternative check:
Quick check using distributive multiplication: (3x - 5xy)(3x - 5xy) gives 9x^2 from 3x*3x, -15x^2 y twice from cross terms totaling -30x^2 y, and +25x^2 y^2 from (-5xy)(-5xy). Same result.


Why Other Options Are Wrong:

-15 term options: they forget there are two cross terms. Swapped coefficients: they confuse (3x)^2 with (5xy)^2. +30 option: sign error; cross term must be negative in (a - b)^2.


Common Pitfalls:
Forgetting the factor 2 in -2ab, or mishandling x*x as x^2, or losing the negative sign in the cross term.


Final Answer:
9x^2 - 30x^2*y + 25x^2*y^2

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