Let-p-q-and-r-be-the-distinct-roots-of-the-polynomial-x-3-22x-2-80x-67-There-exist-real-number-A-B-and-C-such-that-1-s-3-22s-2-80s-67-A-s-p-B-s-q-C-s-r-for-all-real-numbers-

Question Number 135566 by bemath last updated on 14/Mar/21

L e t p, q a n d r b e t h e d i s t i n c t r o o t s

o f t h e p o l y n o m i a l x^{3} - 22 x^{2} + 80 x - 67 .

T h e r e e x i s t r e a l n u m b e r A, B a n d

C s u c h t h a t \frac{1}{s^{3} - 22 s^{2} + 80 s - 67} =

\frac{A}{s - p} + \frac{B}{s - q} + \frac{C}{s - r} f o r a l l r e a l n u m b e r s

s w i t h s \notin {p, q, r} . W h a t i s

\frac{1}{A} + \frac{1}{B} + \frac{1}{C} ?

(a) 243 (b) 244 (c) 245 (d) 246

(e) 247

Commented by EDWIN88 last updated on 14/Mar/21

very nice

Answered by EDWIN88 last updated on 14/Mar/21

\Leftrightarrow 1 = A (s - q) ((s - r) + B (s - p) (s - r) + C (s - p) (s - q)

(∙) s = p \Rightarrow 1 = A (p - q) (p - r); \frac{1}{A} = (p - q) (p - r)

similarly \Rightarrow {\begin{cases} \frac{1}{B} = (q - p) (q - r) \\ \frac{1}{C} = (r - p) (r - q) \end{cases}

\frac{1}{A} + \frac{1}{B} + \frac{1}{C} = p^{2} + q^{2} + r^{2} - pq - pr - qr

= {(p + q + r)}^{2} - 3 (pq + pr + qr)

= 22^{2} - 3 (80) = 244

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