If (x+1/x)^2=3 then show that x^3+1/x^3=0 [ ICSE Class 9 S.Chand (Fundamentals of Mathematics) Chapter 3 Expansion and Factorisation Exercise-3.1 Solution ]
$\left(x + \frac{1}{x}\right)^2$ = 3
$\Rightarrow \left(x + \frac{1}{x}\right)$ =$\pm \sqrt{3}$
$\therefore \left(x^3 + \frac{1}{x^3}\right)$
= $\left(x + \frac{1}{x}\right)^3 – 3.x\ldotp \frac{1}{x}\left(x + \frac{1}{x}\right)$
= $\left(x + \frac{1}{x}\right)^3 – 3\left(x + \frac{1}{x}\right)$
= $\left( \pm \sqrt{3}\right)^3 – 3\left( \pm \sqrt{3}\right)$
=$\left( \pm \sqrt{3} \times \sqrt{3} \times \sqrt{3}\right) – \left( \pm 3\sqrt{3}\right)$
= $\left( \pm 3\sqrt{3}\right) – \left( \pm 3\sqrt{3}\right)$
= 0
$\therefore \left(x^3 + \frac{1}{x^3}\right)$ =0 [Proved]