How to find the roots of the following polynomial. $$\frac{\alpha_1}{x + \alpha_1} + \frac{\alpha_2}{x + \alpha_2} + \cdots +\frac{\alpha_k}{x + \alpha_k} = 1$$ where $\alpha_i$s are complex numbers.

For the case $k=2$, I get the roots $\pm \sqrt{\alpha_1\alpha_2}$. I was trying to use Vieta's formulas to simplify the expression, but I am unable to do make any progress. Kindly share your thoughts. Thank you.

Also, is there any theoretical significance of these polynomials? Kindly share some references. Thanks again.