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Which of the subsequent is equivalent to the polynomial $x − 2 y 3 + 2 y − 3 z 3 + 3 z − x 3$ by using the algebraic identities?

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3 x + 2 y 2 y − 3 z 3 z − x

3 x − 2 y 2 y − 3 z 3 z − x

3 x + 2 y 2 y + 3 z 3 z − x

3 x − 2 y 2 y − 3 z 3 z + x

answer is B.

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Detailed Solution

Given the polynomial,

x − 2 y 3 + 2 y − 3 z 3 + 3 z − x 3

We know the algebraic identity,

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

a + b + c = 0,

then

a 3 + b 3 + c 3 = 3 a b c .

Let’s assume

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

Here, we see that,

a + b + c = 0

i . e . x − 2 y + 2 y − 3 z + 3 z − x = 0

Therefore, applying the above algebraic identity, we get,

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

Therefore, the factorization of the polynomial

x − 2 y 3 + 2 y − 3 z 3 + 3 z − x 3

3 x − 2 y 2 y − 3 z 3 z − x .

Hence, option (2) is correct.

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