1.
a) x (y - x)² - x² + 2xy - y² b) x (x - y)² - y (x - y)² + xy² - x²y
= x (y - x)² - (x² - 2xy + y²) = (x - y) (x - y)² + xy (y - x)
= x (y - x)² - (x - y)² = (x - y)³ - xy (x - y)
= x (y - x)² + (y - x)² = (x - y) {(x - y)² - xy)
= (x + 1) (y - x)² = (x - y) (x² - 2xy + y² - xy)
= (x - y) (x² - 3xy + y²)
c) (3x + 1)² - (3x - 1)² d) (x + y)² - (x - y)²
= (3x + 1 - 3x + 1) (3x + 1 + 3x - 1) = (x + y - x + y) (x + y + x - y)
= 12x = 2y . 2x = 4xy
e) (x + y)³ - (x - y)³
= (x + y - x + y) (x² + 2xy + y² + x² - xy + xy - y² + x² - 2xy + y²)
= 2y (3x² + y²) = 6x²y + 2y³
f) x³ + y³ + z³ - 3xyz
= (x + y)³ - 3xy (x + y) + z³ - 3xyz
= (x + y)³ + z³ - 3xy (x + y + z)
= (x + y + z)³ - 3 (x + y + z) (x + y) z - 3xy (x + y + z)
= (x + y + z)³ - 3 (x + y + z) (xy + yz + zx)
= (x + y + z) [(x + y + z)² - 3xy - 3yz - 3zx)]
= (x + y + z) (x² + y² + z² - xy - yz - zx)
