Roots of Unity Questions

Question

Let $\ w = e^{\frac{4\pi i}{7}}$. Evaluate $(2+w)(2+w^2)(2+w^3)(2+w^4)(2+w^5)(2+w^6)$.

How would you evaluate the expression using the concepts of Roots of Unity or other methods?

Answer 1

If $w$ is a primitive $n$-th root of unity then $$x^n-1=(x-1)(x-w)(x-w^2)\cdots(x-w^{n-1}).\tag1$$ Here $w$ is a primitive seventh root of unity, and you seek $$(2+w)(2+w^2)\cdots(2+w^6).\tag2$$ In $(2)$ you have six factors, not seven as in $(1)$ and plus rather than minus signs, but there is a value of $x$ you can choose in $(1)$ to give you something close enough to $(2)$.

Answer 2

$$w^7=1$$

Clearly, the roots of $$x^7-1=0$$ are $w^r,0\le r\le6$

Let $2+x=y\implies0=(y-2)^7-1=y^7-\binom71y^62^1+\cdots-(2^7+1)$

$$\implies\prod_{r=0}^6(2+w^r)=\dfrac{2^7+1}1$$

What is $2+w^0=?$