Factorize each of the following expressions:$ a^{3}-\frac{1}{a^{3}}-2 a+\frac{2}{a} $

Given:

\( a^{3}-\frac{1}{a^{3}}-2 a+\frac{2}{a} \)

To do:

We have to factorize the given expression.

Solution:

We know that,

$a^3 + b^3 = (a + b) (a^2 - ab + b^2)$

$a^3 - b^3 = (a - b) (a^2 + ab + b^2)$

Therefore,

$a^{3}-\frac{1}{a^{3}}-2 a+\frac{2}{a}=(a-\frac{1}{a})(a^{2}+1+\frac{1}{a^{2}})-2(a-\frac{1}{a})$

$=(a-\frac{1}{a})(a^{2}+1+\frac{1}{a^{2}}-2)$

$=(a-\frac{1}{a})(a^{2}-1+\frac{1}{a^{2}})$

$=(a-\frac{1}{a})(a^{2}+\frac{1}{a^{2}}-1)$

Hence, $a^{3}-\frac{1}{a^{3}}-2 a+\frac{2}{a}=(a-\frac{1}{a})(a^{2}+\frac{1}{a^{2}}-1)$.

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Updated on: 10-Oct-2022

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