MHB Is there a non-constant polynomial such that....

[karseme]

It is required to determine if there is a non-constant polynomial p with positive coefficients such that function $x \mapsto p(x^2)-p(x)$ is decreasing on $[1,+\infty \rangle$. What should I do here? How should I exactly determine that? What is the right method? My idea was to use somehow the fact that composition of decreasing and increasing function is decreasing etc. But, then again I have a polynomial here which can be of any degree. So, I am not sure if it would be increasing on the whole domain or not, but I guess I can somehow use the fact that all coefficients are positive. What's to be done here?

 

[Opalg]

It is required to determine if there is a non-constant polynomial p with positive coefficients such that function $x \mapsto p(x^2)-p(x)$ is decreasing on $[1,+\infty \rangle$. What should I do here? How should I exactly determine that? What is the right method? My idea was to use somehow the fact that composition of decreasing and increasing function is decreasing etc. But, then again I have a polynomial here which can be of any degree. So, I am not sure if it would be increasing on the whole domain or not, but I guess I can somehow use the fact that all coefficients are positive. What's to be done here?

If the polynomial $p(x)$ has degree $n$ then $p(x^2)$ will have degree $2n$. What does that tell you about the behaviour of $p(x^2)-p(x)$ as $x$ gets large?

 

[karseme]

I would say that as x gets large the value of polynomial gets large and as it converges to the infinity the value converges to the infinity...

 

[topsquark]

I would say that as x gets large the value of polynomial gets large and as it converges to the infinity the value converges to the infinity...

First, it can't "converge to infinity" it diverges.

Second, a detail. I know this almost doesn't need to be mentioned but you need to answer this question: p(x^2) and p(x) both diverge. What happens when you subtract the two divergent quantities: p(x^2) - p(x)? How do you show which is "bigger?"

-Dan