State True or False.
$125 (x - y)^{3} + (5 y - 3 z)^{3} + (3 z - 5 x)^{3} =$ $15 (x - y) (5 y - 3 z) (3 z - 5 x)$

True

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False

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Solution

The correct option is A True
Formula used: if $a + b + c = 0$ , then $a^{3} + b^{3} + c^{3} = 3 a b c$
$125 (x - y)^{3} + (5 y - 3 z)^{3} + (3 z - 5 x)^{3} = (5 x - 5 y)^{3} + (5 y - 3 z)^{3} + (3 z - 5 x)^{3}$
Now, $(5 x - 5 y) + (5 y - 3 z) + (3 z - 5 x) = 0$
$\Rightarrow (5 x - 5 y)^{3} + (5 y - 3 z)^{3} + (3 z - 5 x)^{3} = 3 (5 x - 5 y) (5 y - 3 z) (3 z - 5 x)$
$= 15 (x - y) (5 y - 3 z) (3 z - 5 x)$
$∴ 125 (x - y)^{3} + (5 y - 3 z)^{3} + (3 z - 5 x)^{3} = 15 (x - y) (5 y - 3 z) (3 z - 5 x)$

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