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

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The discussion centers on finding the values of A, B, and C in the context of the polynomial equation x^3 - 22x^2 + 80x - 67, whose roots are p, q, and r. The equation is expressed in terms of partial fractions, leading to the relationship between the coefficients and the roots. Participants analyze the structure of the polynomial and apply techniques from algebra to derive the values of A, B, and C. Ultimately, the question posed is whether the sum of the reciprocals, 1/A + 1/B + 1/C, equals 244. The conclusion is that the correct 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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