Question

Factorize x⁴ + x² y² + y⁴

Answer 1

Adding x² y² and subtracting x² y² to the given equation.

= x⁴ + x² y² + y⁴ + x² y² - x² y²

= x⁴ + 2x² y² + y⁴ - x² y²

= (x² )²  + 2 x x² x y²  + ( y² )²  - ( xy)²

Using identity a² + 2ab + b²  = (a + b)²

= (x²  + y² )²  - ( xy)²

Using identity a² - b² = (a + b)(a - b)

= (x²  + y²  + xy)(x²  + y²  - xy )

∴ x⁴  + x² y²  + y⁴   = (x²  + y²  + xy)(x²  + y²  - xy)