Verify that - x3-y3-z3-3 x y z-1-2x-y-z-x-y2-y-z2-z-x2-

We need to verify that x^3 + y^3 + z^3 3xyz = 1/2(x+y+z)[(x y)^2 + (y z)^2 + (z x)^2)] R.H.S= 1/2(x+y+z)[(x y)^2 + (y z)^2 + (z x)^2)] = 1/2(x+y+z ...

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Byju's Answer

Standard IX

Mathematics

Algebraic Identity 3

Verify that ....

Question

Verify that : $x^{3} + y^{3} + z^{3} - 3 x y z = \frac{1}{2} (x + y + z) [{(x - y)}^{2} + {(y - z)}^{2} + {(z - x)}^{2})]$

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Solution

We need to verify that $x^{3} + y^{3} + z^{3} - 3 x y z = \frac{1}{2} (x + y + z) [{(x - y)}^{2} + {(y - z)}^{2} + {(z - x)}^{2})]$

R.H.S= $\frac{1}{2} (x + y + z) [{(x - y)}^{2} + {(y - z)}^{2} + {(z - x)}^{2})]$

= $\frac{1}{2} (x + y + z) [x^{2} + y^{2} - 2 x y + y^{2} + z^{2} - 2 y z + z^{2} + x^{2} - 2 x z]$

= $\frac{1}{2} (x + y + z) [2 x^{2} + 2 y^{2} + 2 z^{2} - 2 x y - 2 y z - 2 x z]$

= $\frac{1}{2} (x + y + z) 2 (x^{2} + y^{2} + z^{2} - x y - y z - x z)$

= $(x + y + z) 2 (x^{2} + y^{2} + z^{2} - x y - y z - x z)$ =L.H.S because it is an identity.

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Verify that : x3+y3+z3−3xyz=12(x+y+z)[(x−y)2+(y−z)2+(z−x)2)]

Verify that : $x^{3} + y^{3} + z^{3} - 3 x y z = \frac{1}{2} (x + y + z) [{(x - y)}^{2} + {(y - z)}^{2} + {(z - x)}^{2})]$