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Binomial Expression

Write the fol...

Question

Write the following in the expanded form:

(viii) $(\frac{x}{y} + \frac{y}{z} + \frac{z}{x})^{2}$

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Solution

Given, $(\frac{x}{y} + \frac{y}{z} + \frac{z}{x})^{2}$
Using Identity,
$(a + b + c)^{2} = a^{2} + b^{2} + c^{2} + 2 a b + 2 b c + 2 c a$
$∴ (\frac{x}{y} + \frac{y}{z} + \frac{z}{x})^{2}$
$= (\frac{x}{y})^{2} + (\frac{y}{z})^{2} + (\frac{z}{x})^{2} + 2 \times \frac{x}{y} \times \frac{y}{z} + 2 \times \frac{y}{z} \times \frac{z}{x} + 2 \times \frac{x}{y} \times \frac{z}{x}$
$= \frac{x^{2}}{y^{2}} + \frac{y^{2}}{z^{2}} + \frac{z^{2}}{x^{2}} + 2 \times \frac{x}{z} + 2 \times \frac{y}{x} + 2 \times \frac{z}{y}$
$= \frac{x^{2}}{y^{2}} + \frac{y^{2}}{z^{2}} + \frac{z^{2}}{x^{2}} + \frac{2 x}{z} + \frac{2 y}{x} + \frac{2 z}{y}$

Hence, $(\frac{x}{y} + \frac{y}{z} + \frac{z}{x})^{2} = \frac{x^{2}}{y^{2}} + \frac{y^{2}}{z^{2}} + \frac{z^{2}}{x^{2}} + \frac{2 x}{z} + \frac{2 y}{x} + \frac{2 z}{y} .$

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Binomial Expression

Standard XII Mathematics

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