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Partial Fraction Decomposition

Source: 2019 AMC 10A #24

February 8, 2019
algebrapartial fractionsAMCAMC 10AMC 10 A2019 AMC 10A

Problem Statement

Let pp, qq, and rr be the distinct roots of the polynomial x3−22x2+80x−67x^3 - 22x^2 + 80x - 67. It is given that there exist real numbers AA, BB, and CC such that 1s3−22s2+80s−67=As−p+Bs−q+Cs−r\dfrac{1}{s^3 - 22s^2 + 80s - 67} = \dfrac{A}{s-p} + \dfrac{B}{s-q} + \frac{C}{s-r} for all s∉{p,q,r}s\not\in\{p,q,r\}. What is 1A+1B+1C\tfrac1A+\tfrac1B+\tfrac1C?
<spanclass=′latex−bold′>(A)</span>243<spanclass=′latex−bold′>(B)</span>244<spanclass=′latex−bold′>(C)</span>245<spanclass=′latex−bold′>(D)</span>246<spanclass=′latex−bold′>(E)</span>247<span class='latex-bold'>(A) </span>243\qquad<span class='latex-bold'>(B) </span>244\qquad<span class='latex-bold'>(C) </span>245\qquad<span class='latex-bold'>(D) </span>246\qquad<span class='latex-bold'>(E) </span> 247
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