Multiply this partial fraction by

Multiply this partial fraction by $x-a$. Then let $x\to a$:
$$\frac{1}{Q(x)}=\frac{A}{x-a}+\cdots .$$
Show that $A=1/Q^{\prime }(a)$. When $x=a$ is a double root this fails because $Q^{\prime }(a)=$ ____.

This exercise has fill-in blanks from its source book and no answer key.

Gilbert Strang, Calculus, 3rd edition (Wellesley-Cambridge Press; MIT OpenCourseWare RES.18-001) · §7.4 Partial Fractions, #29