Question 102
Let $n$ be a positive integer and $a_1, \dots, a_n$ be positive real numbers. Let $g(x)$ denote the product $(x + a_1) \cdots (x + a_n)$. Let $a_0$ be a real number and let
$$
f(x) = (x - a_0)g(x) = x^{n+1} + b_1 x^n + b_2 x^{n-1} + \dots + b_n x + b_{n+1}.
$$
Prove that all the coefficients $b_1, b_2, \dots, b_{n+1}$ of the polynomial $f(x)$ are negative if and only if
$a_0 \gt a_1 + a_2 + \dots + a_n.$
$$
f(x) = (x - a_0)g(x) = x^{n+1} + b_1 x^n + b_2 x^{n-1} + \dots + b_n x + b_{n+1}.
$$
Prove that all the coefficients $b_1, b_2, \dots, b_{n+1}$ of the polynomial $f(x)$ are negative if and only if
$a_0 \gt a_1 + a_2 + \dots + a_n.$