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Find the following products:$ \left(3+\frac{5}{x}\right)\left(9-\frac{15}{x}+\frac{25}{x^{2}}\right) $
Given:
\( \left(3+\frac{5}{x}\right)\left(9-\frac{15}{x}+\frac{25}{x^{2}}\right) \)
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,
$(3+\frac{5}{x})(9-\frac{15}{x}+\frac{25}{x^{2}})=(3+\frac{5}{x})[(3)^{2}-3 \times \frac{5}{x}+(\frac{5}{x})^{2}]$
$=(3)^{3}+(\frac{5}{x})^{3}$
$=27+\frac{125}{x^{3}}$
Hence, $(3+\frac{5}{x})(9-\frac{15}{x}+\frac{25}{x^{2}})=27+\frac{125}{x^{3}}$.
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