Factorise : \((x^2 + y^2 -z^2)^2 - 4x^2y^2\)

1.	Factorise : \((x^2 + y^2 -z^2)^2 - 4x^2y^2\)
Answer» \((x^2 + y^2 -z^2)^2 - 4x^2y^2\) = \((x^2 + y^2 -z^2)^2 - (2xy)^2\) = (x² + y² + z² + 2xy)(x² + y² -z² -2xy) [∴ a² - b² = (a-b)(a+b)] = (x² + 2xy + y² - z²)(x² - 2xy + y² - z²) = [(x + y)² -z²] [(x + y)² -z²] = (x + y + z)(x + y - z)(x - y + z)(x - y - z).

Answer»

\((x^2 + y^2 -z^2)^2 - 4x^2y^2\) = \((x^2 + y^2 -z^2)^2 - (2xy)^2\)

= (x² + y² + z² + 2xy)(x² + y² -z² -2xy) [∴ a² - b² = (a-b)(a+b)]

= (x² + 2xy + y² - z²)(x² - 2xy + y² - z²)

= [(x + y)² -z²] [(x + y)² -z²]

= (x + y + z)(x + y - z)(x - y + z)(x - y - z).

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