Write the following in the expanded form: ( x y + y z + z x )^2

Question

Write the following in the expanded form: $\Big(\frac{\text{x}}{\text{y}}+\frac{\text{y}}{\text{z}}+\frac{\text{z}}{\text{x}}\Big)^2$

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Answer

We have, $\Big(\frac{\text{x}}{\text{y}}+\frac{\text{y}}{\text{z}}+\frac{\text{z}}{\text{x}}\Big)^2$
$=\Big(\frac{\text{x}}{\text{y}}\Big)^2+\Big(\frac{\text{y}}{\text{z}}\Big)+\Big(\frac{\text{z}}{\text{x}}\Big)^2$
$+2\frac{\text{x}}{\text{y}}\frac{\text{y}}{\text{z}}+2\frac{\text{y}}{\text{z}}\frac{\text{z}}{\text{x}}+2\frac{\text{z}}{\text{x}}\frac{\text{x}}{\text{y}}$
$\big[\therefore\big(\text{x}+\text{y}+\text{z}\big)$
$=\text{x}^2+\text{y}^2+\text{z}^2+2\text{xy}+2\text{yz}+2\text{xz}\big]$
$\therefore\Big(\frac{\text{x}}{\text{y}}+\frac{\text{y}}{\text{z}}+\frac{\text{z}}{\text{x}}\Big)^2$
$=\Big(\frac{\text{x}^2}{\text{y}^2}\Big)+\Big(\frac{\text{y}^2}{\text{z}^2}\Big)+\Big(\frac{\text{z}^2}{\text{x}^2}\Big)+2\frac{\text{x}}{\text{z}}+2\frac{\text{y}}{\text{x}}+2\frac{\text{x}}{\text{y}}$

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