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Verify that x^3+y^3+z^3-3xyz=1/2(x+y+z)[(x-y)^2+(y-z)^2+(z-x)^2] - Mathematics

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Verify that `x^3+y^3+z^3-3xyz=1/2(x+y+z)[(x-y)^2+(y-z)^2+(z-x)^2]`

 

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Solution

It is known that,

x3 + y3 + z3 - 3xyz = (x + y + z)(x2 + y2 + z2 - xy - yz - zx)

`therefore x^3+y^3+z^3-3xyz=1/2(x+y+z)[(x-y)^2+(y-z)^2+(z-x)^2]`

                                     `= 1/2(x+y+z)[2x^2+2y^2+2z^2-2xy-2yz-2zx]`

                                     `= 1/2(x+y+z)[(x^2+y^2-2xy)+(y^2+z^2-2yz)+(x^2+z^2-2zx)]`

                                     `= 1/2(x+y+z)[(x-y)^2+(y-z)^2+(z-x)^2]`

 

Concept: Algebraic Identities
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APPEARS IN

NCERT Class 9 Maths
Chapter 2 Polynomials
Exercise 2.5 | Q 12 | Page 49

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