Consider the following two polynomials:

$$ g=x^3 - x^2 - (c + 2)x + c $$

and

$$ h=x^3 - x^2 - cx + c $$

The roots of $h$ are $1$ and $\pm \sqrt{c}$. I am interested in obtaining the roots of $g$, using the fact that $g=h-2x$.

Numerical calculation indicates that the roots of $g$ are indeed not very far from those of $h$. For example, the largest root of $g$ is $\sqrt{c}+\Theta(\frac{1}{\sqrt{c}})$.

In this example, I can derive such formulas using elementary methods but I wonder if there is a principled way to derive them.