Consider the following polynomial: $p(x)=x^{3}-(k-1)x^{2}-(2k-1)x+(k-1)^{2}$, where $k \geq 5$ is a fixed parameter. I am trying to find a strong lower bound on the largest root $x\_{\max}$ of the polynomial of the form $x\_{\max} \geq f(k)$.


So far I was able to show that $x\_{\max} \geq k$ which is quite close to the actual value of the root but I need some help going the extra mile.



Bounds on the largest root of a polynomial
I would write this as a comment, but as I'm new here, it doesn't allow me to do so.


Anyway, one can make your bound better by replacing $x\_{max}\ge k$ by $x\_{max}\ge k+\frac{1}{2k}$. This can be seen by considering $q(y)=p(y+k)=y^3+(2k+1)y^2+(k^2+1)y-k+1=0$. Definitely $q(\frac{1}{k})>0$ and actually, it's not difficult to see $q(\frac{1}{2k})<0$ for $k>3$.


I guess you can keep doing this by adding negative degrees of $k$ to your expression and make your bound better and better. Hope this will help you.


