The sum of $162^{ ext {th }}$ power of the roots of the equation $x^{3}-2 x^{2}+2 x-1=0$ is

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The sum of $162^{	ext {th }}$ power of the roots of the equation $x^{3}-2 x^{2}+2 x-1=0$ is

Professor PC Thomas Classes · Answer

(3): We have, $x^{3}-2 x^{2}+2 x-1=0$ Since, $x=1$ satisfies the given equation.\ $	herefore x-1$ is a factor of (i)\ $	herefore\left(x^{3}-2 x^{2}+2 x-1ight)=(x-1)\left(x^{2}-x+1ight)$\ $	herefore \quad x=1, \frac{1 \pm i \sqrt{3}}{2}$ i.e., $x=1,-\omega^{2},-\omega$\ $	herefore \quad$ Sum of $162^{	ext {th }}$ power of roots\ $=(1)^{162}+\left(-\omega^{2}ight)^{162}+(-\omega)^{162}$\ $=1+\omega^{324}+\omega^{162}=1+\left(\omega^{3}ight)^{108}+\left(\omega^{3}ight)^{54}$\ $=1+1+1=3$.\