Question:`A = x - 2, B = x^2 + 2x + 4, C = x^2 - 8` তিনটি বীজগাণিতিক রাশি।

 ক. যোগফল নির্ণয় কর: `a/(bc) + b/(ca) + c/(ab) + (a - b)/(ac)`

 খ. সরল কর: `1/A xx (x - 2)/B + (6x)/C`

 গ. প্রমাণ কর যে, `1/A xx (x + 2) -: (x + 2)/C = 1` 

Answer ক. নির্ণেয় যোগফল 

       `= a/(bc) + b/(ca) + c/(ab) + (a - b)/(ac)`

       `= (a^2 + b^2 + c^2 + b(a - b))/(abc)`

       `= (a^2 + b^2 + c^2 + ab - b^2)/(abc)`

       `= (a^2 + ab + c^2)/(abc)` (Ans)

 খ. দেওয়া আছে,

              A = x - 2

              B =` x^2 + 2x + 4`

              C =` x^3 - 8`

                 =` x^3 - 2^3`

                 =` (x - 2) (x^2 + 2x + 2^2)`

                 = `(x - 2) (x^2 + 2x + 4)`

  প্রদত্ত রাশি,`= 1/A xx (x - 2)/B + (6x)/C`

                =` 1/(x - 2) xx 1/(x^2 + 2x + 4) + (6x)/((x - 2) (x^2 + 2x + 4))`

                =` 1/(x^2 + 2x + 4) + (6x)/((x - 2)(x^2 + 2x + 4))`

                =` (x - 2 + 6x)/((x - 2)(x^2 + 2x + 4))`

                =` (7x - 2)/(x^3 - 8)` (Ans)


  গ. বামপক্ষ `= 1/A xx (x + 2)/B -: (x + 2)/C`

               `= 1/(x - 2) xx (x + 2)/(x^2 + 2x + 4) -: (x + 2)/(x^3 - 8)`

               `= (x + 2)/((x - 2) (x^2 + 2x + 4) xx (x^3 - 8)/(x + 2)`

               `= (x^3 - 8)/(x^3 - 2^3)`

               `= (x^3 - 8)/(x^3 - 8)`

                 = 1

                 = ডানপক্ষ

             `:. 1/A xx (x + 2)/B -: (x + 2)/C = 1`  (প্রমাণিত) 

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