`(a - b)/(a^2b^2), (b - c)/(b^2c^2), (c - d)/(c^2a^2)`

  প্রদত্ত ভগ্নাংশগুলোর হর `a^2b^2, b^2c^2 ও c^2a^2` 

      এর ল,সা,গু `= a^2b^2c^2`

   হরগুলোর ল,সা,গু/প্রথম ভগ্নাংশের হর `(a^2b^2c^2)/(a^2b^2) = c^2`

  `:. (a - b)/(a^2b^2) = (c^2(a - b))/(a^2b^2c^2)`

   হরগুলোর ল,সা,গু/দ্বিতীয় ভগ্নাংশের হর `= (a^2b^2c^2)/(b^2c^2) = a^2`

   `:. (b - c)/(b^2c^2) = (a^2(b - c))/(a^2b^2c^2)`

   এবং হরগুলোর ল,সা,গু/তৃতীয় ভগ্নাংশের হর `= (a^2b^2c^2)/(c^2a^2) = b^2`

   `:. (c - d)/(c^2a^2) = (b^2(c - d))/(a^2b^2c^2)`

   নির্ণেয় সাধারণ হরবিশিষ্ট ভগ্নাংশগুলো হলো

   `(c^2(a - b))/(a^2b^2c^2), (a^2(b - c))/(a^2b^2c^2) ও (b^2(c - d))/(a^2b^2c^2)`

  উত্তর:  `(c^2(a - b))/(a^2b^2c^2), (a^2(b - c))/(a^2b^2c^2) ও (b^2(c - d))/(a^2b^2c^2)`

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

   প্রদত্ত ভগ্নাংশগুলোর হর x + y, y + z ও z + x

    এর ল,সা,গু = (x + y) (y + z) (z + x)

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

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

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

   হরগুলোর ল,সা,গু/দ্বিতীয় ভগ্নাংশের হর

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

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

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

    এবং হরগুলোর ল,সা,গু/তৃতীয় ভগ্নাংশের হর

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

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

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

    :. সাধারণ হরবিশিষ্ট ভগ্নাংশগুলো হলো,

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

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

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

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

ক. `(a - b)/a + (a + b)/b`

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

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

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

    উত্তর: `(a^2 + 2ab - b^2)/(ab)`



   খ. `a/(bc) + b/(ca) + c/(ab)`

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

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

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

           `= (yz(x - y) + zx(y - z) + xy(z - x))/(xyz)`

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

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

   উত্তর:  `= (3xyz - x^2y - y^2z - z^2x)/(xyz)`

   ঘ.  `(x + y)/(x - y) + (x - y)/(x + y)`

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

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

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

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

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

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

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

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

                                 `+ 1/(x(x - 4) - 1(x - 4))`

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

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

 `= (x^2 - 7x + 12 + x^2 - 6x + 8 + x^2 - 5x + 6)/((x - 1)(x - 2)(x - 3)(x - 4))`

  উত্তর: `= (3x^2 - 18x + 26)/((x - 1)(x - 2)(x - 3)(x - 4))`

`1/(a^2 - b^2) + 1/(a^2 + ab + b^2) + 1/(a^2 - ab + b^2)`

  `= 1/(a^2 - b^2) + (a^2 - ab + b^2 + a^2 + ab + b^2)/((a^2 + ab + b^2)(a^2 - ab + b^2)`

  `= 1/(a^2 - b^2) + (2a^2 + 2b^2)/((a^2 + b^2 + ab)(a^2 + b^2 - ab))`

  ` = 1/(a^2 - b^2) + (2a^2 + 2b^2)/((a^2 + b^2)^2 - (ab)^2)`

  ` = 1/(a^2 - b^2) + (2a^2 + 2b^2)/(a^4 + 2a^2b^2 + b^4 - a^2b^2)`

   `= 1/(a^2 - b^2) + (2a^2 + 2b^2)/(a^4 + a^2b^2 + b^4)`

   `= (a^4 + a^2b^2 + b^4 + 2(a^2 + b^2)(a^2 - b^2))/((a^2 - b^2)(a^4 + a^2b^2 + b^4))`

    `= (a^4 + a^2b^2 + b^4 + 2(a^4 - b^4))/((a^2 - b^2)(a^4 + a^2b^2 + b^4))`

    `= (a^4 + a^2b^2 + b^4 + 2a^4 - 2b^4)/((a^2 - b^2) {(a^2)^2 + a^2b^2 + (b^2)^2})`

   `= (3a^4 + a^2b^2 - b^4)/((a^2)^3 - (b^2)^3)`

   `= (3a^4 + a^2b^2 - b^4)/(a^6 - b^6)`

   উত্তর: `(3a^4 + a^2b^2 - b^4)/(a^6 - b^6)`