Question 93
The coefficients of the polynomial
$$
P(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0
$$
are real numbers such that $a_i = a_{n-i}$ for every $i = 0, 1, \dots, n$, and $a_n \neq 0$. Let $x_1, x_2, \dots, x_k$ be all the real roots of the polynomial $P(x)$ without repetitions.
a. Prove that
$$
|x_1| + |x_2| + \dots + |x_k| \ge k.
$$
b. Is the inequality definitely strict in the case $k \gt 1$?
$$
P(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0
$$
are real numbers such that $a_i = a_{n-i}$ for every $i = 0, 1, \dots, n$, and $a_n \neq 0$. Let $x_1, x_2, \dots, x_k$ be all the real roots of the polynomial $P(x)$ without repetitions.
a. Prove that
$$
|x_1| + |x_2| + \dots + |x_k| \ge k.
$$
b. Is the inequality definitely strict in the case $k \gt 1$?