Question 1196220
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Find A, B and C if {{{(A/(x-1)) + (B/(x-2)) + (C/(x-3)) = (2x^2 - 6x + 6)/((x-1)(x-2)(x-3))}}}.
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<pre>
(1)  Consider both sides at x--> 1.

     Left side tends to infinity as {{{A/(x-1)}}}. 

     Right side tends to infinity as {{{(2*1^2-6*1+6)/((x-1)*(1-2)*(1-3))}}} = {{{2/(2*(x-1))}}} = {{{1/(x-1)}}}.

     From it, we conclude that A = 1.




(2)  Consider both sides at x--> 2.

     Left side tends to infinity as {{{B/(x-2)}}}. 

     Right side tends to infinity as {{{(2*2^2-6*2+6)/((2-1)*(x-2)*(2-3))}}} = {{{2/((-1)*(x-2))}}} = {{{-2/(x-2)}}}.

     From it, we conclude that B = -2.




(3)  Consider both sides at x--> 3.

     Left side tends to infinity as {{{C/(x-3)}}}. 

     Right side tends to infinity as {{{(2*3^2-6*3+6)/((3-1)*(3-2)*(x-3))}}} = {{{6/(2*1*(x-3))}}} = {{{3/(x-3)}}}.

     From it, we conclude that C = 3.



<U>ANSWER</U>.  A = 1,  B = -2,  C = 3.
</pre>

Solved.


It is the way to solve the problem practically "on fingers", 
without reducing to the system of 3 equations in 3 unknowns.