Factorise:

$\dfrac{x}{y} + \dfrac{y}{x} + \dfrac{y}{z} + \dfrac{z}{y} + \dfrac{x}{z} + \dfrac{z}{x} + 3$

Given,

$\dfrac{x}{y} + \dfrac{y}{x} + \dfrac{y}{z} + \dfrac{z}{y} + \dfrac{x}{z} + \dfrac{z}{x} + 3$

Multiplying the given expression by xyz,

= xyz × $\Big(\dfrac{x}{y} + \dfrac{y}{x} + \dfrac{y}{z} + \dfrac{z}{y} + \dfrac{x}{z} + \dfrac{z}{x} + 3)$

= x²z + y²z + y²x + z²x + x²y + z²y + 3xyz

= x²y + xy² + y²z + yz² + z²x + zx² + 3xyz

= (x + y + z)(xy + yz + zx)

Divide by xyz

= $\dfrac{(x + y + z)(xy + yz + zx)}{xyz}$

= $(x + y + z) \times \dfrac{xy + yz + zx}{xyz}$

= (x + y + z) $\Big(\dfrac{1}{z} + \dfrac{1}{x} + \dfrac{1}{y}\Big)$

Hence, $\dfrac{x}{y} + \dfrac{y}{x} + \dfrac{y}{z} + \dfrac{z}{y} + \dfrac{x}{z} + \dfrac{z}{x} + 3 = (x + y + z)\Big(\dfrac{1}{x} + \dfrac{1}{y} + \dfrac{1}{z}\Big)$.