Let $A ( X ) , B ( X )$ be non-zero polynomials in $\mathbb { C } [ X ]$ such that $0 \leq \operatorname { deg } A \leq \operatorname { deg } B - 2$ and $A ( X )$ and $B ( X )$ do not share any roots. Let $\alpha _ { 1 } , \alpha _ { 2 } , \ldots , \alpha _ { k }$ be the roots of $B ( X )$. Suppose that each of them is a simple root.
Show that
$$\sum _ { j = 1 } ^ { k } \frac { A \left( \alpha _ { j } \right) } { B ^ { \prime } \left( \alpha _ { j } \right) } = 0$$