Hi,

I'm not sure if you can help me with this, but I'm currently looking for an upper bound on the real part of the roots of a polynomial with real coefficients. In other words, I have a polynomial

$a_n x^n + a_{n - 1} x^{n - 1} + ... + a_1 x + a_0 = 0$

where $a_0, a_1, \dots, a_{n}$ are real coefficients. Suppose that the roots are $t_1, t_2, \dots, t_n$. I want to find an bound $t^*$ such that $Re(t_i) \leq t^*$ for $i = 1, \dots, n$. Is there a theorem that provides such a result? I've seen Rouche's theorem, but from what I've read, this only tells me if the root lies in some circle of specified size. I may be missing something but I don't think that it tells me much about the real part of the root.

Any help is appreciated. Thank You.