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Factorize the following expressions:

`a^3 - 1/a^3 - 2a + 2/a`

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#### Solution

`= (a^3 - 1/a^3) - 2(a - 1/a)`

`= (a^3 - (1/a)^3) - 2(a - 1/a)`

`= (a - 1/a)(a^2 + a xx 1/a + (1/a)^2) - 2(a - 1/a)` `[∵ a^3 - b^3 = (a - b)(a^2 + ab + b^2)]`

`= (a - 1/a)(a^2 + 1 + 1/a^2) - 2(a - 1/a)`

`= (a - 1/a)(a^2 + 1 + 1/a^2 - 2)`

`=(a - 1/a)(a^2 + 1 + -1)`

`∴ a^3 - 1/a^3 - 2a + 2/a = (a- 1/a)(a^2 + 1/a^2 - 1)`

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