I have a polynomial, given for parameters $x$ in $\mathbb{R}_+$ and $\alpha$ and $\beta$ in $\mathbb{R}_+^{n}$ by :

$$P(t) = \sum\limits_{i=1}^n \left\{\left(\beta_i - \frac{\alpha_i}{x} - t\right) \prod\limits_{j = 1, j \neq i}^n (\beta_j - t)\right\}$$

How can i find roots, in function of $\alpha,\beta,x$ ? I have troubles factorising it further..

Edit : As a coment pointed out, since my parameter are all non-zero, this is the same a looking for roots of :

$$\sum\limits_{i=1}^n \frac{\alpha_i}{\beta_i - t} = x$$

6more comments