Basic Analysis I — Real Analysis · Problem 3.3.6

Continuous Functions — Jiri Lebl

Mathematics — Real Analysis · extreme-and-intermediate-value-theorems

Problem

Suppose

g(x)

is a monic polynomial of positive even degree

d

, that is,

g(x) = x^d + b_{d-1}x^{d-1} + \cdots + b_1 x + b_0,

for some real numbers

b_0, b_1, \ldots, b_{d-1}

. Suppose

g(0) < 0

. Show that

g

has at least two distinct real roots.