Simplify each of the following expressions:$ (x^{2}-x+1)^{2}-(x^{2}+x+1)^{2} $

Given:

\( (x^{2}-x+1)^{2}-(x^{2}+x+1)^{2} \)

To do:

We have to simplify \( (x^{2}-x+1)^{2}-(x^{2}+x+1)^{2} \).

Solution:

We know that,

$(a+b+c)^2=a^2+b^2+c^2+2ab+2bc+2ca$

Therefore,

$(x^{2}-x+1)^{2}-(x^{2}+x+1)^{2}=(x^{4}+x^{2}+1-2 x^{3}-2 x+2 x^{2})-(x^{4}+x^{2}+1+2 x^{3}+2 x+2 x^{2})$

$=x^{4}+x^{2}+1-2 x^{3}-2 x+2 x^{2}-x^{4}-x^{2}-1-2 x^{3}-2 x-2 x^{2}$

$=-4 x^{3}-4 x$

$=-4 x(x^{2}+1)$

Hence, $(x^{2}-x+1)^{2}-(x^{2}+x+1)^{2}=-4 x(x^{2}+1)$.

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Updated on: 10-Oct-2022

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