Roots of Polynomial: Find $\frac{1}{A}+\frac{1}{B}+\frac{1}{C}$

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The discussion centers on the polynomial $x^3 - 22x^2 + 80x - 67$ and its roots $p$, $q$, and $r$. It establishes a relationship involving the constants $A$, $B$, and $C$ in the equation $\frac{1}{s^3 - 22s^2 + 80s - 67} = \frac{A}{s-p} + \frac{B}{s-q} + \frac{C}{s-r}$ for all $s \notin \{p, q, r\}$. The objective is to determine the value of $\frac{1}{A} + \frac{1}{B} + \frac{1}{C}$, with the proposed answer being 244. The correct evaluation of this expression confirms that the answer is indeed 244.

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Let $p,\,q$ and $r$ be the distinct roots of the polynomial $x^3-22x^2+80x-67$. It is given that there exist real numbers $A,\,B$ and $C$ such that

$\dfrac{1}{s^3-22s^2+80s-67}=\dfrac{A}{s-p}+\dfrac{B}{s-q}+\dfrac{C}{s-r}$ for all $s\not \in \{p,\,q,\,r\}$. What is $\dfrac{1}{A}+\dfrac{1}{B}+\dfrac{1}{C}$?
 
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Is answer 244?
 

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