Proving that $|\Phi_n(x)| > x-1$

Question

Let $\Phi_n$ be the n-th cyclotomic polynomial.

I'd like to prove that $$\forall n \geq 2, \forall x \in [2, \infty[, |\Phi_n(x)| > x-1$$

The result is clear when $n$ is prime, but I'm struggling to prove the general thing.

Answer 1

We have $$\tag1\Phi_n(x)=\prod_{1\le k<n\atop\gcd(k,n)=1}(x-\zeta^k)$$ where the $\zeta$ is a primitive $n$th root of unity. If $x>1$ then $|x-\zeta^k|\ge x-1$ for all $k$ because $|\zeta^k|=1$. Thus if $x>2$ we can estimate the absolute value of one factor in $(1)$ as $\ge x-1$ and all others as $\ge x-1\ge 1$. Equality can only hold if $\zeta^k=1$ for all $k$, but that means $n<2$.