Question

The value of the expression $(\omega -1)(\omega -{\omega }^2)(\omega -{\omega }^3)...(\omega -{\omega }^{n-1}),$ where $\omega$ is the nth root of unity, is

• A. $n{\omega }^{n-1}$
• B. $n{\omega }^n$
• C. $(n-1){\omega }^n$
• D. $(n-1){\omega }^{n-1}$

Answer 1

The correct option is A $n{\omega }^{n-1}$

We have, $x^n-1=(x-1)(x-\omega )(x-{\omega }^2)...(x-{\omega }^{n-1})$

$\Rightarrow \frac{x^n-1}{x-\omega }=(x-1)(x-{\omega }^2)...(x-{\omega }^{n-1})$

Putting $x=\omega$ on both sides, we have

$(\omega -1)(\omega -{\omega }^2)...(\omega -{\omega }^{n-1})=\underset{x\to \omega }{lim}\frac{x^n-1}{x-\omega }\left(\frac{0}{0}form\right)$

$=\underset{x\to \omega }{lim}\frac{n{x}^{n-1}}{1}=n{\omega }^{n-1}$