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Find the following products:$ \left(x^{3}+1\right)\left(x^{6}-x^{3}+1\right) $
Given:
\( \left(x^{3}+1\right)\left(x^{6}-x^{3}+1\right) \)
To do:
We have to find the given product.
Solution:
We know that,
$a^{3}+b^{3}=(a+b)(a^{2}-a b+b^{2})$
$a^{3}-b^{3}=(a-b)(a^{2}+a b+b^{2})$
Therefore,
$(x^{3}+1)(x^{6}-x^{3}+1)=(x^{3}+1)[(x^{3})^{2}-x^{3} \times 1+(1)^{2}]$
$=(x^{3})^{3}+(1)^{3}$
$=x^{9}+1$
Hence, $(x^{3}+1)(x^{6}-x^{3}+1)=x^{9}+1$.
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