Factorize each of the following expressions:$ \frac{1}{27} x^{3}-y^{3}+125 z^{3}+5 x y z $

Given:

\( \frac{1}{27} x^{3}-y^{3}+125 z^{3}+5 x y z \)

To do:

We have to factorize the given expression.

Solution:

We know that,

$a^3 + b^3 + c^3 - 3abc = (a + b + c) (a^2 + b^2 + c^2 - ab - bc - ca)$

Therefore,

$\frac{1}{27} x^{3}-y^{3}+125 z^{3}+5 x y z = (\frac{1}{3} x)^{3}+(-y)^{3}+(5 z)^{3}-3 \times \frac{x}{3} \times(-y) \times 5 z$

$=(\frac{1}{3} x-y+5 z)[(\frac{1}{3} x)^{2}+(-y)^{2}+(5 z)^{2}-\frac{1}{3} x \times(-y)-(-y) \times (5 z)-5 z \times \frac{1}{3} x]$

$=(\frac{1}{3} x-y+5 z)(\frac{1}{9} x^{2}+y^{2}+25 z^{2}+\frac{1}{3} x y+5 y z-\frac{5}{3} z x)$

Hence, $\frac{1}{27} x^{3}-y^{3}+125 z^{3}+5 x y z = (\frac{1}{3} x-y+5 z)(\frac{1}{9} x^{2}+y^{2}+25 z^{2}+\frac{1}{3} x y+5 y z-\frac{5}{3} z x)$.

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Updated on: 10-Oct-2022

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