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Algebraic Identities

Without actua...

Question

Without actually calculating the cubes, find the value of ${(- 12)}^{3} + {(7)}^{3} + {(5)}^{3}$

A

0

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B

-36

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C

-1620

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D

-1260

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Solution

The correct option is D
-1260

$x^{3} + y^{3} + z^{3} - 3 x y z = (x + y + z) (x^{2} + y^{2} + z^{2} - x y - y z - z x)$

When $(x + y + z) = 0$

$x^{3} + y^{3} + z^{3} = 3 x y z$

$- 12 + 7 + 5 = 0$

$∴ {(- 12)}^{3} + {(7)}^{3} + {(5)}^{3} = 3 (- 12) (7) (5) = - 1260$

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