Q:

Calculus 1 polynomials and derivatives

Need your help with a Calculus 1 exercise. 
Let $P,Q$ be polyomials with $deg(P)<deg(Q)$ and $Q(x)=(x-a_1)(x-a_2)\cdots (x-a_n)$ where $a_i \neq a_j$ for $1 \leq i,j \leq n$ and $i \neq j$. Show that if we write $$\frac{P}{Q}=\frac{A_1}{(x-a_1)}+\frac{A_2}{(x-a_2)}+ \cdots + \frac{A_n}{(x-a_n)}$$ then $A_j=P(a_j)/Q'(a_j)$.

A:

Hint
See that
$$P(x)= \frac{A_1Q(x)}{x-a_1}+h(x)$$
But 
$$P(a_1)=A_1(a_1-a_2)(a_1-a_3)...(a_1-a_n)+h(a_1)$$
But is easy to see that $h(a_1)=0$ and we also have $(a_1-a_2)(a_1-a_3)...(a_1-a_n)=Q'(a_1)$.
Can you finish?

