Question:
Find the value
$a^{3}-1 / a^{3}-2 a+2 a$
Solution:
$=\left(a^{3}-1 / a^{3}\right)-2(a-1 / a)$
$=\left(a^{3}-(1 / a)^{3}\right)-2(a-1 / a)$
$=(a-1 / a)\left(a^{2}+a \times 1 / a+(1 / a)^{2}\right)-2(a-1 / a)$
$\left[\therefore a^{3}-b^{3}=(a-b)\left(a^{2}+a b+b^{2}\right)\right]$
$=(a-1 / a)\left(a^{2}+1+1 / a^{2}\right)-2(a-1 / a)$
$=(a-1 / a)\left(a^{2}+1+1 / a^{2}-2\right)$
$=(a-1 / a)\left(a^{2}+1 / a^{2}-1\right)$
$\therefore a^{3}-1 / a^{3}-2 a+2 a=(a-1 / a)\left(a^{2}+1 / a^{2}-1\right)$
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