Zeros of polynomial

Asked Jun 22 '20 at 08:06

Active Jun 22 '20 at 08:19

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I am struggling with the following last part of a problem. I have reduced everything to the following lemma which I can not prove. Let

$$P(z) = a_{n} x^{n} + \cdots + a_{0}$$

a polynomial with real coefficients such that $0 \leq a_{0} \leq \cdots \leq a_{n}$. How can I prove that all the roots of $P(z)$ are contained in the circumference $|z|\leq 1$?

Thank you all for your time.

edited Jun 22 '20 at 08:19

Rodrigo de Azevedo

asked Jun 22 '20 at 08:06

Samantha Smith

1

Are there more conditions? For example $P = x^2+x$ gives a contradiction – Benjamin Wang Jun 22 '20 at 08:10
Sorry, It was $|z|\leq 1$. – Samantha Smith Jun 22 '20 at 08:11
If the coefficients are strictly decreasing then $|z| < 1$ holds, compare https://math.stackexchange.com/q/3536695/42969. – Martin R Jun 22 '20 at 08:21
I think the same proof that is written in the post can be applied to this case. Don't you think @MartinR? – Samantha Smith Jun 22 '20 at 08:23

Zeros of polynomial

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