272. Упростите выражение: а)

Решение:

Приведём дроби к общему знаменателю \( (x-y)^2 \) и \( (y-x)^2 = (x-y)^2 \).

• \( \frac{x^3+y^3}{(x-y)^2} + \frac{3xy^2-y^3}{(y-x)^2} = \frac{x^3+y^3}{(x-y)^2} + \frac{3xy^2-y^3}{(x-y)^2} = \frac{x^3+3xy^2}{(x-y)^2} \)
• \( \frac{3x^2y+y^3}{2xy-x^2-y^2} = \frac{y(3x^2+y^2)}{-(x^2-2xy+y^2)} = \frac{y(3x^2+y^2)}{-(x-y)^2} \)
• \( \frac{x^3+3xy^2}{(x-y)^2} + \frac{y(3x^2+y^2)}{-(x-y)^2} = \frac{x^3+3xy^2-3x^2y-y^3}{(x-y)^2} \)
• \( x^3-y^3 = (x-y)(x^2+xy+y^2) \)
• \( 3xy^2-3x^2y = 3xy(y-x) \)
• \( \frac{(x-y)(x^2+xy+y^2) + 3xy(y-x)}{(x-y)^2} = \frac{(x-y)(x^2+xy+y^2) - 3xy(x-y)}{(x-y)^2} \)
• \( \frac{(x-y)(x^2+xy+y^2-3xy)}{(x-y)^2} = \frac{x^2-2xy+y^2}{x-y} = \frac{(x-y)^2}{x-y} = x-y \)

Ответ: x-y.