" 21."2xy-(x^(2)+y^(2)-z^(2)

Factorize: 2xy-(x^(2)+y^(2)-z^(2))

Prove that |(x^(2),x^(2)-(y-z)^(2),yz),(y^(2),y^(2)-(z-x)^(2),zx),(z^(2),z^(2)-(x-y)^(2),xy)|=(x-y)(y-z)(z-x)(x+y+z)(x^(2) + y^(2) + z^(2)) .

Prove that quad det ([yx-x^(2),zx-y^(2),xy-z^(2)zx-y^(2),xy-z^(2),yz-x^(2)xy-z^(2),yz-x^(2),zx-y^(2)]) is divisible by (x+y+z) and hence find the quotient.

Simplify: (x+y-2z)^(2)-x^(2)-y^(2)-3z^(2)+4xy

Find the following produts: (i) (x+y+2z)(x^(2)+y^(2)+4z^(2)-xy-2yz-2xz)(2x-y+2z)(4x^(2)+y^(2)+9z^(2)+2xy+3yz-6xz)

Prove that |{:(x^(2),,x^(2)-(y-z)^(2),,yz),(y^(2),,y^(2)-(z-x)^(2),,zx),(z^(2),,z^(2)-(x-y)^(2),,xy):}| =(x-y) (y-z) (z-x)(x+y+z) (x^(2)+y^(2)+z^(2))

Prove that |{:(x^(2),,x^(2)-(y-z)^(2),,yz),(y^(2),,y^(2)-(z-x)^(2),,zx),(z^(2),,z^(2)-(x-y)^(2),,xy):}| =(x-y) (y-z) (z-x)(x+y+z) (x^(2)+y^(2)+z^(2))

(x-y-z)^(2)-(x^(2)+y^(2)+z^(2))=2(yz-zx-xy)