Question

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

Answer 1

Answer:

step by step explanation :-

Verify that :-

x³+y³+z³-3xyz= 1/2(x+y+z)[(x-y)²+(y-z)²+(z-x)²]

Solution :-

L. H. S=x³+y³+z³-3xyz

We know that,

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

=1/2 (x+y+z) [ 2x²+2y²+2z²-2xy-2yz-2zx]

=1/2 (x+y+z) [ x²+x²+y²+y²+z²+z²-2xy-2yz-2zx]

=1/2 (x+y+z) [ x²+y²-2xy+y²+z²-2yz+z²+x²-2zx]

=1/2 (x+y+z) [ (x²+y²-2xy)+(y²+z²-2yz)+(z²+x²-2zx]

=1/2 (x+y+z) [ (x-y)²+(y-z)²+(z-x)²]

L.H. S. =R. H. S.

Hence Verified.

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