Question:`A = x^2 - 5x + 6, B = x^2 - 7x + 12`

 `C = x^2 - 9x + 20` তিনটি বীজগাণিতিক রাশি।

  ক. `x/y` থেকে `(x + y)/y` এর বিয়োগ কর।

  খ. `1/B + 1/C` কে লঘিষ্ট আকারে প্রকাশ কর।

  গ. `1/A, 1/B, 1/C` কে সাধারণ হর বিশিষ্ট ভগ্নাংশে প্রকাশ কর। 

Answer ক. নির্ণেয় বিয়োগফল `= x/y - (x + y)/y`

                       `= (x - x - y)/y`

                       `= (-y)/y`

                         = - 1

              উত্তর: - 1

 
  খ. দেওয়া আছে, 

          `B = x^2 - 7x + 12` এবং `C = x^2 - 9x + 20`

        `:. 1/B + 1/C = 1/(x^2 - 7x + 12) + 1/(x^2 - 9x + 20)`

        `= 1/(x^2 - 4x - 3x + 12) +  1/(x^2 - 5x - 4x + 20)`

        `= 1/(x(x - 4) - 3(x - 4)) + 1/(x(x - 5) - 4(x - 5))`

        `= 1/((x - 3) (x - 4)) + 1/((x - 4)(x - 5))`

        `= (x - 5 + x - 3)/((x - 3)(x - 4)(x - 5))`

        `= (2x - 8)/((x - 3)(x - 4)(x - 5))`  (Ans)

  
  গ. `1/A = 1/(x^2 - 5x + 6)`

      `= 1/(x^2 - 3x - 2x + 6)`

      `= 1/(x(x - 3) - 2(x - 3))`

      `= 1/((x - 2) (x - 3))`

    `1/B = 1/(x^2 - 7x + 12)`

          `= 1/(x^2 - 4x - 3x + 12)`

          `= 1/(x(x - 4) - 3(x - 4))`

          `= 1/((x - 4)(x - 3))`

    এবং `1/C = 1/(x^2 - 9x + 20)`

              `= 1/(x^2 - 4x - 5x + 20)`

              `= 1/(x(x - 4) - 5(x - 4))`

              `= 1/((x - 4) (x - 5))`

         প্রদত্ত ভগ্নাংশগুলোর হর `x^2 - 5x + 6, x^2 - 7x + 12` ও `x^2 - 9x + 20` 

        এর ল,সা,গু হলো (x - 2) (x - 3) (x - 4) (x - 5)

        এখন, হরগুলোর ল,সা,গু/প্রথম ভগ্নাংশের হর

         `= ((x - 2) (x - 3) (x - 4) (x - 5))/((x - 2) (x - 3))`

         `= (x - 4) (x - 5)`

      `:. 1/A = (1 xx (x - 4) (x - 5))/((x - 2)(x - 3)(x - 4) (x - 5))`

         `= ((x - 4)(x - 5))/((x - 2) (x - 3) (x - 4) (x - 5))`

           হরগুলোর ল,সা,গু/দ্বিতীয় ভগ্নাংশের হর


        `= ((x - 2) (x - 3) (x - 4) (x - 5))/((x - 3) (x - 4))`

        `= (x - 2) (x - 5)`

     `:. 1/B = (1 xx (x - 2) (x - 5))/((x - 3) (x - 4) (x - 2) (x - 5))`

        `= ((x - 2) (x - 5))/((x - 2) (x - 3) (x - 4) (x - 5))`

      এবং হরগুলোর ল,সা,গু/তৃতীয় ভগ্নাংশের হর

          `= ((x - 2)(x - 3)(x - 4)(x - 5))/((x - 4) (x - 5))`

          `= (x - 2) (x - 3)`

      `:. 1/C = (1 xx (x - 2) (x - 3))/((x - 4)(x - 5) xx (x - 2) (x - 3))`

               `= ((x - 2) (x - 3))/((x - 2) (x - 3)(x - 4)(x - 5))`

         :. নির্ণেয় সাধারণ হরবিশিষ্ট ভগ্নাংশগুলো হলো:

          `((x - 4) (x - 5))/((x - 2) (x - 3) (x - 4) (x - 5))`

          `((x - 2) (x - 5))/((x - 2) (x - 3) (x - 4) (x - 5))`

          `((x - 2) (x - 3))/((x - 2) (x - 3)(x - 4)(x - 5))` (Ans) 

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