Find the following products: $ \left(\frac{3}{x}-2 x^{2}\right)\left(\frac{9}{x^{2}}+4 x^{4}-6 x\right) $

Given: 

\( \left(\frac{3}{x}-2 x^{2}\right)\left(\frac{9}{x^{2}}+4 x^{4}-6 x\right) \)

To do: 

We have to find the given product.

Solution: 

We know that,

$a^{3}+b^{3}=(a+b)(a^{2}-a b+b^{2})$

$a^{3}-b^{3}=(a-b)(a^{2}+a b+b^{2})$

Therefore,

$(\frac{3}{x}-2 x^{2})(\frac{9}{x^{2}}+4 x^{4}-6 x)=(\frac{3}{x}-2 x^{2})(\frac{9}{x^{2}}-6 x+4 x^{4})$

$=(\frac{3}{x}-2 x^{2})[(\frac{3}{x})^{2}-\frac{3}{x} \times 2 x^{2}+(2 x^{2})^{2}]$

$=(\frac{3}{x})^{3}-(2 x^{2})^{3}$

$=\frac{27}{x^{3}}-8 x^{6}$

 Hence, $(\frac{3}{x}-2 x^{2})(\frac{9}{x^{2}}+4 x^{4}-6 x)=\frac{27}{x^{3}}-8 x^{6}$.

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Updated on: 10-Oct-2022

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