Try this beautiful Problem on Algebra based on Roots of Polynomial from AMC 10 A, 2019. You may use sequential hints to solve the problem.
Let $p, q,$ and $r$ be the distinct roots of the polynomial $x^{3}-22 x^{2}+80 x-67$. It is given that there exist real numbers $A, B$, and $C$ such that $\frac{1}{s^{3}-22 s^{2}+80 s-67}=\frac{A}{s-p}+\frac{B}{s-q}+\frac{C}{s-r}$
for all $s \notin{p, q, r} .$ What is $\frac{1}{A}+\frac{1}{B}+\frac{1}{C} ?$
,
Algebra
Linear Equation
Pre College Mathematics
AMC-10A, 2019 Problem-24
$244$
The given equation is $\frac{1}{s^{3}-22 s^{2}+80 s-67}=\frac{A}{s-p}+\frac{B}{s-q}+\frac{C}{s-r}$.....................(1)
If we multiply both sides we will get
Multiplying both sides by $(s-p)(s-q)(s-r)$ we will get
$$
1=A(s-q)(s-r)+B(s-p)(s-r)+C(s-p)(s-q)
$$
Now can you finish the problem?
Now Put $S=P$ we will get $\frac{1}{A}=(p-q)(p-r)$............(2)
Now Put $S=q$ we will get $\frac{1}{B}=(q-p)(q-r)$...........(3)
Now Put $S=r$ we will get $\frac{1}{C}=(r-p)(r-q)$...........(4)
Now Can you finish the Problem?
Adding (2) +(3)+(4) we get,$\frac{1}{A}+\frac{1}{B}+\frac{1}{C}=p^{2}+q^{2}+r^{2}-p q-q r-p r$
Now Using Vieta's Formulas, $p^{2}+q^{2}+r^{2}=(p+q+r)^{2}-2(p q+q r+p r)=324$ and $p q+q r+p r=80$
Therefore the required answer is $324-80$=$244$
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