Question

p(x) is a polynomial with real coefficients such that for some positive real numbers c,d and for all natural numbers n, we have

c |n|^3 <= |p(n)| <= d |n|^3

Prove p(x) has a real zero

Answer 1

I have the solution, its a little one, but i don;t understand that :|

Answer 2

Please post the solution so that we can have a look too. :)

Answer 3

it simply says,

" Obviously p(x) has degree 3, hence the result. "

o.O
how can we say it has degree 3 like that? :O

Answer 4

If the degree is more than 3 then
\[
\lim_{n\to \infty} \frac {|p(n)|}{n^3} =\infty
\]

Answer 5

If the degree is less than 3 then
\lim_{n\to \infty} \frac {|p(n)|}{n^3} =0

Answer 6

So to satisfy both inequalities, p has to have degree 3

Answer 7

\[\lim_{n\to \infty} \frac {|p(n)|}{n^3} =0

\]
If the degree of p is less than 3