খ.`(x - y)/(xy), (y - z)/(yz), (z - x)/(zx)`

 প্রদত্ত ভগ্নাংশগুলোর হর xy, yz ও zx এর ল, সা,গু = xyz

 এখন হরগুলোর ল.সা.গু/প্রথম ভগ্নাংশের হর

 `= (xyz)/(xy) = z`

 `:. (x - y)/(xy) = ((x - y)z)/(xyz) `

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

  হরগুলোর ল.সা,গু/দ্বিতীয় ভগ্নাংশের হর 

`= (xyz)/(yz) = x`

 `:. (y - z)/(xy) = ((y - z).x)/(yz.x)`

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

 এবং হরগুলোর ল.সা,গু/তৃতীয় ভগ্নাংশের হর 

`= (xyz)/(zx) = y`

 `:. (z - x)/(zx) = ((z - x).y)/(zx.y)`

`= (y.(z - x))/(xyz)`

  :. নির্ণেয় সাধারণ হরবিশিষ্ট ভগ্নাংশগুলো হলো

   `(z(x - y))/(xyz), (x(y - z))/(xyz), (y(z - x))/(xyz)`

   উত্তর: `(z(x - y))/(xyz), (x(y - z))/(xyz), (y(z - x))/(xyz)`

গ.`x/(x - y), y/(x + y), z/(x(x + y))`

  প্রদত্ত ভগ্নাংশগুলোর হর (x - y),(x + y) ও x(x + y)

  এর ল,সা,গু = x(x - y) (x + y)

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

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

`= x(x + y)`

 `:. x/(x - y) = (x.x(x + y))/((x - y).x(x + y))`

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

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

  হরগুলোর ল,সা,গু/দ্বিতীয় ভগ্নাংশের হর 

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

 `= x (x - y)`

`:. y/(x + y) = (y. x(x - y))/((x + y). x(x - y))`

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

                 `= (xy(x - y))/(x(x^2 - y^2))`

  এবং হরগুলোর ল,সা,গু/তৃতীয় ভগ্নাংশের হর 

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

  = x - y

`:. z/(x(x + y)) = (z(x - y))/(x(x + y) (x - y))`

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

 `= (z(x - y))/(x(x^2 - y^2)`

  :. নির্ণেয় সাধারণ হরবিশিষ্ট ভগ্নাংশগুলো হলো

  `(x^2(x + y))/(x(x^2 - y^2)), (xy(x - y))/(x(x^2 - y^2)), (z(x - y))/(x(x^2 - y^2))`

  উত্তর: `(x^2 (x + y))/(x(x^2 - y^2)), (xy(x - y))/(x(x^2 - y^2)) (z(x - y))/(x(x^2 - y^2))`

ঘ.`(x + y)/(x - y)^2, (x - y)/(x^3 + y^3), (y - z)/(x^2 - y^2)`

   এখানে, `(x - y)^2 = (x - y) (x - y)`

            `x^3 + y^3 = (x + y) (x^2 - xy + y^2)`

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

    প্রদত্ত ভগ্নাংশগুলোর হর `(x - y)^2, x^3 + y^3 ও  x^2 - y^2`

     এর ল,সা,গু `(x - y) (x - y) (x + y) (x^2 - xy + y^2)`

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

      `((x - y)(x - y) (x + y) (x^2 - xy + y^2))/((x - y) (x - y))`

     `= (x + y) (x^2 - xy + y^2) = x^3 + y^3`

      `:. (x + y)/(x - y)^2 = ((x + y) (x^3 + y^3))/((x - y)^2 (x^3 + y^3))`

     হরগুলোর ল,সা,গু/দ্বিতীয় ভগ্নাংশের হর 

     `((x - y) (x - y) (x + y) (x^2 - xy + y^2))/((x + y) (x^2 - xy + y^2))`

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

       `= (x - y)^2`

     `:. (x - y)/(x^3 + y^3) = ((x - y) (x - y)^2)/((x^3 + y^3) (x - y)^2)`

      `= ((x - y))^3/((x - y)^2 (x^3 + y^3))`

     এবং হরগুলোর ল,সা,গু/তৃতীয় ভগ্নাংশের হর

      `= ((x - y) (x - y) (x + y) (x^2 - xy + y^2))/((x + y) (x - y))`

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

    `:. (y - z)/(x^2 - y^2) = ((y - z) (x - y) (x^2 - xy + y^2))/((x^2 - y^2) (x - y) (x^2 - xy + y^2))`

      `= ((x - y) (y - z) (x^2 - xy + y^2))/((x + y) (x - y) (x - y) (x^2 - xy + y^2))`

      `= ((x - y) (y - z) (x^2 - xy + y^2))/((x - y)^2 (x^3 + y^3))`

      :. সাধারণ হরবিশিষ্ট ভগ্নাংশগুলো হলো

   `((x + y) (x^3 + y^3))/((x - y)^2 (x^3 + y^3)), ((x - y)^3)/((x - y)^2 (x^3 + y^3))`

                                         `((x - y) (y - z) (x^2 - xy + y^2))/((x - y)^2 (x^3 + y^3))`


   উত্তর:  `((x + y) (x^3 + y^3))/((x - y)^2 (x^3 + y^3)), ((x - y)^3)/((x - y)^2 (x^3 + y^3))`

                                         `((x - y) (y - z) (x^2 - xy + y^2))/((x - y)^2 (x^3 + y^3))`

`a/(a^3 + b^3), b/(a^2 + ab + b^2), c/(a^3 - b^3)`

এখানে, `a^3 + b^3 = (a + b) (a^2 - ab + b^2)`

`a^2 + ab + b^2 = a^2 + ab + b^2`

`a^3 - b^3 = (a - b) (a^2 + ab + b^2)`

 প্রদত্ত ভগ্নাংশগুলোর হর `a^3 + b^3, a^2 + ab + b^2 ও a^3 - b^3`

 এর ল,সা,গু `= (a + b) (a^2 - ab + b^2) (a - b) (a^2 + ab + b^2)`

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

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

 `= (a - b) (a^2 + ab + b^2) = a^3 - b^3`

`:. a/(a^3 + b^3) = (a(a^3 - b^3))/((a^3 + b^3) (a^3 - b^3)) `

`= (a(a^3 - b^3))/(a^6 - b^6)`

 হরগুলোর ল,সা,গু/দ্বিতীয় ভগ্নাংশের হর

 `((a + b) (a^2 - ab + b^2) (a - b) (a^2 + ab + b^2))/(a^2 + ab + b^2)`

`= (a + b) (a - b) (a^2 - ab + b^2)`

`= (a - b) (a^3 + b^3)`

 `:. b/(a^2 + ab + b^2)`

 `= (b(a - b) (a^3 + b^3))/((a - b) (a^3 + b^3) (a^2 + ab + b^2))`

 `= (b(a - b) (a^3 + b^3))/((a^3 - b^3) (a^3 + b^3)) `

 `= (b(a - b)(a^3 + b^3))/(a^6 - b^6)`

  এবং হরগুলোর ল,সা,গু/তৃতীয় ভগ্নাংশের হর

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

 `= a^3 + b^3`

 `:. c/(a^3 - b^3) = (c(a^3 + b^3))/((a^3 - b^3)(a^3 + b^3)) `

 `= (c(a^3 + b^3))/(a^6 - b^6)`

 :. নির্ণেয় সাধারণ হরবিশিষ্ট ভগ্নাংশগুলো হলো

  `(a(a^3 - b^3))/(a^6 - b^6), (b(a - b)(a^3 + b^3))/(a^6 - b^6)  (c(a^3 + b^3))/(a^6 - b^6)`

  উত্তর: `(a(a^3 - b^3))/(a^6 - b^6), (b(a - b)(a^3 + b^3))/(a^6 - b^6)  (c(a^3 + b^3))/(a^6 - b^6)`

`1/(x^2 - 5x + 6), 1/(x^2 - 7x + 12), 1/(x^2 - 9x + 20)`

এখানে, `x^2 - 5x + 6, = x^2 - 3x - 2x + 6`

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

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

`x^2 - 7x + 12 = x^2 - 4x - 3x + 12`

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

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

 `x^2 - 9x + 20 = x^2 - 5x - 4x + 20`

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

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

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

   ও `x^2 - 9x + 20` এর ল,সা,গু

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

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

 `:. 1/(x^2 - 5x + 6)`

 `= (1(x - 5)(x - 4))/((x - 3)(x - 2)(x - 5)(x - 4))`

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

  হর গুলোর ল,সা,গু/দ্বিতীয় ভগ্নাংশের হর

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

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

 `:. 1/(x^2 - 7x + 12) = (1(x - 2)(x - 5))/((x - 4)(x - 3) (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 - 5)(x - 4))`

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

 `:. 1/(x^2 - 9x + 20) = (1(x - 2)(x - 3))/((x - 5)(x - 4)(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)) ও ((x - 2)(x - 3))/((x - 2) (x - 3)(x - 4)(x - 5))`

  উত্তর: `((x - 4)(x - 5))/((x - 2)(x - 3)(x - 4)(x - 5)), ((x - 2)(x - 5))/((x - 2)(x - 3)(x - 4)(x - 5))`
          ও `((x - 2)(x - 3))/((x - 2)(x - 3)(x - 4)(x - 5))`