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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Updated on: 10-Oct-2022

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