I will start my question by providing some necessary context:
Let $g(x) = c_lx^l + ... + a_1x + a$ be a polynomial of degree $l$.
$g$ is said to have no missing coefficients if $c_i \neq 0$ for all $i = 0,...,l$.
The coefficients of $g$ are strictly alternating if $a_0>0, a_1<0, a_2>0,...,$ or $a_0<0, a_1>0, a_2<0,...$
I need to prove the following lemma:
Let $f(x) = a_mx^m + ... + a_1x + a_0$ be a polynomial with coefficients $a_i$ in $\mathbb{Z}$ for $i = 0,...,m$. Suppose that $\alpha_1, ..., \alpha_m$ are roots of f and $\Re e(\alpha_i) < b$ for all $i$ where $b$ is a fixed number. Then the polynomial f(x+b) has no missing coefficients and all the coefficients have the same sign. Deduce that the coefficients of f(-x+b) are strictly alternating.
I have tried to just fill in $x+b$ into the equation $f(x) = a_mx^m + ... + a_1x + a_0$, but this seems to be way too much calculating work for an algebra-question. Maybe I am overlooking something, but I just don't see a better way. Can someone help me with this please?
P.S.: Right before this lemma, I needed to and have been able to prove the following lemma
Let $f \in \mathbb{Z}[x]$. If $\alpha$ is a complex root of $f$, then so is the conjugate $\bar{\alpha}$.
Maybe this is needed to prove the lemma I need to prove.