Find the quotient when (2x³ - 5x²y + 5xy² - 3y³) is divided by (2x - 3y)?
- x² - xy + y²
- x² - xy - y²
- x² + xy + y²
- None of these
Explanation
Step 1: Perform polynomial long division or synthetic division
To find the quotient when (2x³ - 5x²y + 5xy² - 3y³) is divided by (2x - 3y), let's try factoring or direct division.
Step 2: Factor the numerator if possible
Notice that (2x³ - 5x²y + 5xy² - 3y³) can be factored as:
(2x³ - 3x²y) + (-2x²y + 3xy²) + (2xy² - 3y³)
= x²(2x - 3y) - xy(2x - 3y) + y²(2x - 3y)
= (2x - 3y)(x² - xy + y²)
Step 3: Identify the quotient
Given the factorization (2x - 3y)(x² - xy + y²), the quotient when (2x³ - 5x²y + 5xy² - 3y³) is divided by (2x - 3y) is x² - xy + y².
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