How to find the roots of the following polynomial. $$\frac{n_1}{x + n_1} + \frac{n_2}{x + n_2} + \cdots +\frac{n_k}{x + n_k} = 1$$ where $n_i$s are natural numbers.

For the case $k=2$, I get the roots $\pm \sqrt{n_1n_2}$. 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.

1. I was trying induction on $k$ but didn't lead anywhere.
2. I tried to represent it as some indefinite integral. But no success.

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

How to find the solutions $x $ of the following equation: $$\frac{n_1}{x + n_1} + \frac{n_2}{x + n_2} + \cdots +\frac{n_k}{x + n_k} = 1$$ where $n_i$s are natural numbers.

For the case $k=2$, I get the solutions $\pm \sqrt{n_1n_2}$. 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.

1. I was trying induction on $k$ but didn't lead anywhere.
2. I tried to represent it as some indefinite integral. But no success.

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

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.

How to find the roots of the following polynomial. $$\frac{n_1}{x + n_1} + \frac{n_2}{x + n_2} + \cdots +\frac{n_k}{x + n_k} = 1$$ where $n_i$s are natural numbers.

For the case $k=2$, I get the roots $\pm \sqrt{n_1n_2}$. 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.

1. I was trying induction on $k$ but didn't lead anywhere.
2. I tried to represent it as some indefinite integral. But no success.

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