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

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In summary, the conversation discusses determining if there exists a non-constant polynomial $p$ with positive coefficients such that the function $x \mapsto p(x^2)-p(x)$ is decreasing on the interval $[1,+\infty \rangle$. The participants suggest using the fact that the composition of a decreasing and increasing function is decreasing, as well as considering the behavior of the polynomial $p(x)$ and $p(x^2)$ as $x$ gets large. However, it is noted that the polynomial may diverge as $x$ gets large, and it is important to determine which quantity is larger when subtracting them.
  • #1
karseme
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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?
 
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  • #2
karseme said:
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?
 
  • #3
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...
 
  • #4
karseme said:
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
 

FAQ: Is there a non-constant polynomial such that....

Is there a non-constant polynomial such that it has an infinite number of roots?

Yes, there is a non-constant polynomial called the zero polynomial, which has an infinite number of roots. This polynomial is defined as P(x) = 0 for all values of x.

Is there a non-constant polynomial such that it has no real roots?

Yes, there is a non-constant polynomial called the complex polynomial, which has no real roots. This polynomial is defined as P(x) = x2 + 1, where x is a real number.

Is there a non-constant polynomial such that it has only one root?

Yes, there is a non-constant polynomial called the linear polynomial, which has only one root. This polynomial is defined as P(x) = ax + b, where a and b are constants and a ≠ 0.

Is there a non-constant polynomial such that it has no roots?

Yes, there is a non-constant polynomial called the constant polynomial, which has no roots. This polynomial is defined as P(x) = c, where c is a constant and c ≠ 0.

Is there a non-constant polynomial such that it has roots in the form of complex numbers?

Yes, there are many non-constant polynomials that have roots in the form of complex numbers. Some examples include P(x) = x2 + 4x + 5 and P(x) = x3 - 1.

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