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Which of the options is equivalent to the following polynomial?

$− z + x − 2 y x 2 + 4 y 2 + z 2 + 2 x y + x z − 2 y z$

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x 3 − 8 y 3 − z 3 − 6 x y z

x 3 + 8 y 3 + z 3 − 6 x y z

x 3 + 8 y 3 + z 3 + 6 x y z

x 3 − 8 y 3 − z 3 + 6 x y z

answer is A.

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Detailed Solution

The given polynomial is,

− z + x − 2 y x 2 + 4 y 2 + z 2 + 2 x y + x z − 2 y z

Simplify the above polynomial.

∴ − z + x − 2 y x 2 + 4 y 2 + z 2 + 2 x y + x z − 2 y z = x + − 2 y + − z x 2 + − 2 y 2 + − z 2 − x − 2 y − − z − 2 y − − z x

Apply the algebraic identity,

a 3 + b 3 + c − 3 a b c = a + b + c a 2 + b 2 + c 2 − a b − b c − c a

⇒ − z + x − 2 y x 2 + 4 y 2 + z 2 + 2 x y + x z − 2 y z = x + − 2 y + − z x 2 + − 2 y 2 + − z 2 − x − 2 y − − z − 2 y − − z x

= x 3 + − 2 y 3 + − z 3 − 3 x − 2 y − z = x 3 − 8 y 3 − z 3 − 6 x y z

Therefore, the product of the polynomial is,

− z + x − 2 y x 2 + 4 y 2 + z 2 + 2 x y + x z − 2 y z = x 3 − 8 y 3 − z 3 − 6 x y z .

Hence, option (1) is correct.

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