Deriving an identity related to partial fractions of large products

Solution 1:

The Heaviside Method for Partial Fractions requires that the numerator have a lower degree than the denominator. However, a small modification fixes things up:

The Heaviside Method for Partial Fractions gives $$ \begin{align} \frac{\prod\limits_{k=1}^n(x-a_k)}{\prod\limits_{k=1}^n(x-b_k)} &=1+\frac{\prod\limits_{k=1}^n(x-a_k)-\prod\limits_{k=1}^n(x-b_k)}{\prod\limits_{k=1}^n(x-b_k)}\\ &=1+\sum_{k=1}^n\frac1{x-b_k}\frac{\prod\limits_{j=1}^n(b_k-a_j)\color{#AAA}{-\prod\limits_{j=1}^n(b_k-b_j)}}{\prod\limits_{\substack{j=1\\j\ne k}}^n(b_k-b_j)}\\[3pt] &=1+\sum_{k=1}^n\frac{b_k-a_k}{x-b_k}\prod_{\substack{j=1\\j\ne k}}^n\frac{b_k-a_j}{b_k-b_j} \end{align} $$ where the grayed-out term is $0$.