Question

factorise using suitable identities
$( \frac{a}{b} ) {}^{3} +( \frac{b}{c} ) {}^{3} + ( \frac{c}{a} ) {}^{3} - 3$
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Answer 1

Answer:

$(\frac{a}{b} +\frac{b}{c} + \frac{c}{a}) (\frac{a^2}{b^2} + \frac{b^2}{c^2} + \frac{c^2}{a^2} - \frac{a}{c} - \frac{b}{a} - \frac{c}{b})$

Step-by-step explanation:

Using Identity x³ + y³ + z³ - 3xyz = (x+y+z)(x²+y²+z²-xy-yz-xz)

We can use this Identity as multiplication of first 3 terms cancels each other i.e. = 1

$(\frac{a}{b})^3 + (\frac{b}{c})^3 + (\frac{c}{a})^3 -3\\= (\frac{a}{b} +\frac{b}{c} + \frac{c}{a}) (\frac{a^2}{b^2} + \frac{b^2}{c^2} + \frac{c^2}{a^2} - \frac{a.b}{b.c} - \frac{b.c}{c.a} - \frac{c.a}{a.b})\\= (\frac{a}{b} +\frac{b}{c} + \frac{c}{a}) (\frac{a^2}{b^2} + \frac{b^2}{c^2} + \frac{c^2}{a^2} - \frac{a}{c} - \frac{b}{a} - \frac{c}{b})$