KnowledgeBoat Logo

Mathematics

Factorise :

(x2+y2−z2)2−4x2y2(x^2 + y^2 - z^2)^2 - 4x^2y^2

Factorisation

1 Like

Answer

(x2+y2−z2)2−4x2y2=(x2+y2−z2)2−(2xy)2=((x2+y2−z2)−2xy)((x2+y2−z2)+2xy)=(x2+y2−z2−2xy)(x2+y2−z2+2xy)=(x2+y2−2xy−z2)(x2+y2+2xy−z2)=((x−y)2−z2)((x+y)2−z2)=((x−y)−z)((x−y)+z)((x+y)−z)((x+y)+z)=(x−y−z)(x−y+z)(x+y−z)(x+y+z)(x^2 + y^2 - z^2)^2 - 4x^2y^2\\[1em] = (x^2 + y^2 - z^2)^2 - (2xy)^2\\[1em] = \Big((x^2 + y^2 - z^2) - 2xy\Big)\Big((x^2 + y^2 - z^2) + 2xy\Big)\\[1em] = (x^2 + y^2 - z^2 - 2xy)(x^2 + y^2 - z^2 + 2xy)\\[1em] = (x^2 + y^2 - 2xy - z^2)(x^2 + y^2 + 2xy - z^2)\\[1em] = ((x - y)^2 - z^2)((x + y)^2 - z^2)\\[1em] = ((x - y) - z)((x - y) + z)((x + y) - z)((x + y) + z)\\[1em] = (x - y - z)(x - y + z)(x + y - z)(x + y + z)

Hence, (x2+y2−z2)2−4x2y2=(x−y−z)(x−y+z)(x+y−z)(x+y+z)(x^2 + y^2 - z^2)^2 - 4x^2y^2 = (x - y - z)(x - y + z)(x + y - z)(x + y + z).

Answered By

2 Likes

Related Questions