Write the Following in Expanded Form
\[ \left(\frac{x}{y} + \frac{y}{z} + \frac{z}{x}\right)^2 \]
Solution:
Using identity:
\[ (a+b+c)^2 = a^2+b^2+c^2+2ab+2bc+2ca \]
\[ \left(\frac{x}{y} + \frac{y}{z} + \frac{z}{x}\right)^2 \]
\[ = \left(\frac{x}{y}\right)^2 + \left(\frac{y}{z}\right)^2 + \left(\frac{z}{x}\right)^2 + 2\left(\frac{x}{y}\right)\left(\frac{y}{z}\right) + 2\left(\frac{y}{z}\right)\left(\frac{z}{x}\right) + 2\left(\frac{z}{x}\right)\left(\frac{x}{y}\right) \]
\[ = \frac{x^2}{y^2} + \frac{y^2}{z^2} + \frac{z^2}{x^2} + \frac{2x}{z} + \frac{2y}{x} + \frac{2z}{y} \]