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Is the given statement true or false?

${(3 a - 2 b)}^{3} + {(2 b - 5 c)}^{3} + {(5 c - 3 a)}^{3} = 3 (3 a - 2 b) (2 b - 5 c) (5 c - 3 a) .$

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False

True

answer is A.

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Detailed Solution

The given polynomial is

{(3 a - 2 b)}^{3} + {(2 b - 5 c)}^{3} + {(5 c - 3 a)}^{3} .

We know that

(x^{3} + y^{3} + z^{3}) = 3 x y z

x + y + z = 0.

Consider here,

x = 3 a - 2 b, y = 2 b - 5 c and z = 5 c - 3 a .

\begin{array}{l} \Rightarrow x + y + z = 3 a - 2 b + 2 b - 5 c + 5 c - 3 a = 0 \\ \Rightarrow x + y + z = 0 \end{array}

Therefore we can use the above formula.

\Rightarrow {(3 a - 2 b)}^{3} + {(2 b - 5 c)}^{3} + {(5 c - 3 a)}^{3} = 3 (3 a - 2 b) (2 b - 5 c) (5 c - 3 a)

Thus, the given equation is true, and so option (1) is correct.

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