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Question 12Ve...

Question

Question 12
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

$x^{3} + y^{3} + z^{3} - 3 x y z = \frac{1}{2} (x + y + z) [(x - y)^{2} + (y - z)^{2} + (z - x)^{2}] R . H . S . = \frac{1}{2} (x + y + z) [x^{2} + y^{2} - 2 x y + y^{2} + z^{2} - 2 y z + z^{2} + x^{2} - 2 z x] = \frac{1}{2} (x + y + z) [2 x^{2} + 2 y^{2} + 2 z^{2} - 2 x y - 2 y z - 2 z x] = (x + y + z) [x^{2} + y^{2} + z^{2} - x y - y z - z x] = x [x^{2} + y^{2} + z^{2} - x y - y z - z x] + y (x^{2} + y^{2} + z^{2} - x y - y z - z x) + z (x^{2} + y^{2} + z^{2} - x y - y z - z x) = x^{3} + x y^{2} + x z^{2} - x^{2} y - x y z - z x^{2} + y x^{2} + y^{3} + y z^{2} - x y^{2} - y^{2} z - z x y + z x^{2} + z y^{2} + z^{3} - z x y - y z^{2} - z^{2} x x^{3} + y^{3} + z^{3} - 3 x y z = L . H . S .$

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Q. Question 12
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}]

Q. State True or False.

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}]

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}]

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}]

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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