Question 1145006
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            Surely and certainly,  this problem is advanced and is intended for advanced students. 


            So,  I will assume that your level corresponds to the problem's level.



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(1)  {{{1/(a^2 - a + 1)}}} = {{{(a+1)/(a^3+1)}}}.


     It follows from the standard identity  {{{a^3+1}}} = {{{(a+1)*(a^2-a+1)}}}.


     Further, since "a" is the root of the equation  {{{x^3+4}}} = 0,  we have  


         {{{a^3+4}}} = 0;  hence,   {{{a^3+1}}} = {{{(a^3+4)-3}}} = {{{0-3}}} = -3.   


     Therefore,  {{{1/(a^2 - a + 1)}}} = {{{(a+1)/(a^3+1)}}} = {{{-(a+1)/3}}}.    (1)



(2)  Similarly,  {{{1/(b^2 - b + 1)}}} = {{{-(b+1)/3}}}.             (2)


(3)  Therefore,


         k = - 3[(1/(a² - a + 1) + (1/(b² - b + 1)] + c  = -3*( {{{-(a+1)/3 - (b+1)/3}}} ) + c = (a+1) + (b+1) + c = (a + b + c) + 2.


      The sum (a + b + c) is the coefficient of the given polynomial  p(x) = x^3 + 4  at "x"; so, it is 0 (zero, ZERO) :

          a + b + c = 0.


      Therefore,  k = 0 + 2 = 2.     <U>ANSWER</U>


<U>ANSWER</U>.  k = 2.
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It is how the given problem &nbsp;<U>is assumed to be solved</U>  &nbsp;and  how it &nbsp;<U>should be solved</U>.