Prove that  |(yz-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. - Mathematics

Prove that  |(yz-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.

Solution

Let |(yz-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)|

Applying C1C1+C2+C3 , we get

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

=(xy+yz+zxx2y2z2)|(1,zx-y^2,xy-z^2),(1,xy-z^2,yz-x^2),(1,yz-x^2,zx-y^2)|

Applying R2R2R1 and R3R3R1, we get

=(xy+yz+zxx2y2z2)|(1,zx-y^2,xy-z^2),(0,(x+y+z)(y-z),(x+y+z)(z-x)),(0,(x+y+z)(y-x),(x+y+z)(z-y))|

=(x+y+z)2(xy+yz+zxx2y2z2)|(1,zx-y^2,xy-z^2),(0,(y-z),(z-x)),(0,(y-x),(z-y))|

=(x+y+z)2(xy+yz+zxx2y2z2)[{(yz)(zy)(zx)(yx)}0+0]

=(x+y+z)2(xy+yz+zxx2y2z2)

So, ∆ is divisible by (x + y + z).

The quotient when ∆ is divisible by (x + y + z) is (x+y+z)(xy+yz+zxx2y2z2)

Concept: Elementary Transformations
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