Question 24
Let $p,$ $q,$ and $r$ be the distinct roots of the polynomial $$
x^3 - 22x^2 + 80x - 67.
$$ There exist real numbers $A,$ $B,$ and $C$ such that $$
\begin{gathered} \frac{1}{s^3-22s^2+80s-67}\\ =\frac{A}{s-p}+\frac{B}{s-q}\\ \quad+\frac{C}{s-r} \end{gathered}
$$ for all real numbers $s$ with $s\notin\{p,q,r\}.$ What is $$
\dfrac1A+\dfrac1B+\dfrac1C?
$$
x^3 - 22x^2 + 80x - 67.
$$ There exist real numbers $A,$ $B,$ and $C$ such that $$
\begin{gathered} \frac{1}{s^3-22s^2+80s-67}\\ =\frac{A}{s-p}+\frac{B}{s-q}\\ \quad+\frac{C}{s-r} \end{gathered}
$$ for all real numbers $s$ with $s\notin\{p,q,r\}.$ What is $$
\dfrac1A+\dfrac1B+\dfrac1C?
$$