Suppose $f(x)$ and $g(x)$ are two monic polynomials with real coefficients of the same degree $n$. Moreover, the roots of $f$ are $n$ consecutive integers that are known to us. Similarly, the roots of $g$ are also $n$ consecutive integers that are known to us. If we fix $\delta \in (0,1]$, is there a nice upper bound in terms of the roots and $\delta$ for the largest real root of the polynomial $h(x) = f(x) - (1-\delta)g(x)$?
If it helps, we can also assume that the roots of $f$ are given by $0,-1,\ldots,-(n-1)$ and $g(x) = f(x+c)$ where $c$ is some positive integer. So the roots of $g$ would be given by $-c,-c-1,\ldots,-c-(n-1)$.
(In the falling factorial notation, $f(x) = (x)_n$ and $g(x) = (x+c)_n$)
For $x>n+c$, we have $f(x)>x^n$ and $g(x)<(x+c)^n$.
If we want $f(x)>d g(x)$, then it suffices for $x^n>d (x+c)^n$, so $x>d^{1/n}(x+c)$, so it suffices for $x>c/(1-d^{-1/n})$.
Letting $d=1-\delta$, for $x>max(n+c,c/(1-(1-\delta)^{-1/n}))$, we have that $x$ is not a root, giving a simple upper bound. You can get tighter bounds by being more careful about the factorials.