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Simplify each of the following products:$ (x^{3}-3 x^{2}-x)(x^{2}-3 x+1) $
Given:
\( (x^{3}-3 x^{2}-x)(x^{2}-3 x+1) \)
To do:
We have to simplify the given product.
Solution:
We know that,
$(a+b)^2=a^2+b^2+2ab$
$(a-b)^2=a^2+b^2-2ab$
$(a+b)(a-b)=a^2-b^2$
Therefore,
$(x^{3}-3 x^{2}-x)(x^{2}-3 x+1)=x(x^{2}-3 x-1)(x^{2}-3 x+1)$
$=x[(x^{2}-3 x)-1][(x^{2}-3 x)+1]$
$=x[(x^{2}-3 x)^{2}-(1)^{2}]$
$=x[(x^{2})^{2}-2 \times x^{2} \times 3 x+(3 x)^{2}-1]$
$=x[x^{4}-6 x^{3}+9 x^{2}-1]$
$=x^{5}-6 x^{4}+9 x^{3}-x$
Hence, $(x^{3}-3 x^{2}-x)(x^{2}-3 x+1)=x^{5}-6 x^{4}+9 x^{3}-x$.
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