Can Real Coefficients Affect the Roots of a Polynomial?

[dbr]

Homework Statement

Let $z^n + \sum_{k=0}^{n-1}a_kz^k$ be a polynomial with real coefficients $a_k\in[0,1]$. If $z_0$ is a root, prove that $Re(z_0) < 0$ or $|z_0| < \frac{1+\sqrt{5}}{2}$.

Homework Equations

The Attempt at a Solution

I have attempted to solve this problem by contradiction (i.e. assuming there is a root $z_0$ with $Re(z_0) \geq 0$ and $|z_0| \geq \frac{1+\sqrt{5}}{2}$). I then tried to look for a contradiction in the equations $Re(z_0^n + \sum_{k=0}^{n-1}a_kz_0^k) = 0$ and $Im(z_0^n + \sum_{k=0}^{n-1}a_kz_0^k) = 0$. Unfortunately, I'm not able to find any contradiction.

 

[mighty2000]

Can someone provide some guidance on how to approach this problem?

Thank you for your post. It seems like you are on the right track with your approach. However, instead of looking for a contradiction in the equations Re(z_0^n + \sum_{k=0}^{n-1}a_kz_0^k) = 0 and Im(z_0^n + \sum_{k=0}^{n-1}a_kz_0^k) = 0, try to use the fact that z_0 is a root of the polynomial and the given conditions on the coefficients to derive a contradiction.

Here are some steps that may help you:

1. Write the given polynomial in the form z^n + a_{n-1}z^{n-1} + ... + a_1z + a_0.

2. Use the fact that z_0 is a root of the polynomial to write an equation involving z_0 and the coefficients a_k.

3. Use the given conditions on the coefficients to derive a contradiction. For example, you can use the fact that a_k \in [0,1] to show that the magnitude of z_0 must be less than a certain value.

I hope this helps. Let me know if you need further clarification or assistance. Good luck with your problem!



Scientist