Question

If $1,\omega ,{\omega }^2,\dots ,{\omega }^{n-1}$ are the $n^{\text{th}}$ roots of unity of $x^n=1.$ Then the value of $(5-\omega )(5-{\omega }^2)\dots (5-{\omega }^{n-1})$ is equal to

• A. $1$
• B. $\frac{5^n+1}{4}$
• C. $4^{n-1}$
• D. $\frac{5^n-1}{4}$

Answer 1

The correct option is D $\frac{5^n-1}{4}$

If $1,\omega ,{\omega }^2,\dots ,{\omega }^{n-1}$ are the $n,n^{th}$ roots of unity, then
$x^n-1=(x-1)(x-\omega )(x-{\omega }^2)\dots (x-{\omega }^{n-1})$

Putting $x=5,$ we get
$5^n-1=(5-1)(5-\omega )(5-{\omega }^2)\dots (5-{\omega }^{n-1})$
$\Rightarrow (5-\omega )(5-{\omega }^2)\dots (5-{\omega }^{n-1})=\frac{5^n-1}{4}$