Bounds on polynomial roots

Question

I am looking to find the minimum absolute value of the roots of the following polynomial: $$ux^M - x + 1$$ where $u$ and $M$ are constants. Does a closed form upper or lower bound expression exist?

M is a positive integer.

Answer 1

Consider the substitution $x\mapsto1/z$ to get

$$z^M-z^{M-1}+u$$

and hence the roots $z$ are bounded above by

$$1+\max(1,|u|)$$

and your roots are bounded below by

$$\frac1{1+\max(1,|u|)}$$