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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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