ক. নির্ণেয় যোগফল `= 1/(x - 2) + 1/(x + 2)`

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

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

                      `= (2x)/(x^2 - 4)`   (Ans)

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

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

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

  :. নির্ণেয় যোগফল  `= (x^2 - 4x - 4)/((x + 2) (x^2 + 4)) + (x - 2)/(x^2 - 2x + 4)`

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

` = (x^4 - 2x^3 + 4x^2 - 4x^3 + 8x^2 - 16x - 4x^2 + 8x - 16 + (x^2 - 4) (x^2 + 4))/((x^3 - 2^3) (x^2 + 4))`

 `= (x^4 - 6x^3 + 8x^2 - 8x - 16 + x^4 - 16)/((x^3 - 8)(x^2 + 4))`

 `= (2x^4 - 6x^3 + 8x^2 - 8x - 32)/((x^2 + 4)(x^3 - 8))`  (Ans)

ক. ১ম রাশি `= (2a^2 + ab - b^2)/(a^3 + a^2b - a - b) = (2a^2 + 2ab - ab - b^2)/(a^2 (a + b) - 1(a + b))`

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

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

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

        উত্তর: `(2a - b)/(a^2 - 1)`


 খ. প্রশ্নানুসারে, `(2a - b - c)/(a - b) - {(b - c)^2/((a - b)(a - c)) + (b + c - 2a)/(a - c)}`

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

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

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

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

   `= (4a^2 - 4ab - 4ac + 4bc)/((a - b)(a - c))`

   `= (4(a^2 - ab - ac + bc))/((a - b)(a - c))`

   `= (4{a(a - b) - c(a - b)})/((a - b)(a - c))`

   `= (4(a - b)(a - c))/((a - b)(a - c))`

     = 4

    উত্তর:  4

 
  গ. ৫ম রাশি, 

    `(c - a)/((c - a)(b - c)) + (b - a)/((a - c)(b - c)) + (b - c)/((a - c)(a - b))`

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

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

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

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

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

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

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

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

  `= (- 2(a - b) (a - c))/((a - b)(b - c) (a - c))`

   `= (- 2)/(b - c)`

   `= (- 2)/(- (c - b))`

   `= 2/(c - b)`

      উত্তর:  ` 2/(c - b)`

ক. ১ম রাশি `= (a^4 - b^4)/(a^2 + b^2 - 2ab)`

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

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

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

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

 খ. নির্ণেয় যোগফল

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

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

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

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

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

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

  গ. `(a^4 - b^4)/(a^2 + b^2 - 2ab), ((a + b)^2 - 4ab)/(a^3 - b^3), (a + b)/(a^2 + ab + b^2)`

     এখানে, 

    ১ম ভগ্নাংশের হর `= a^2 + b^2 - 2ab`

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

                      `= (a - b)^2`

                      `= (a - b) (a - b)`

    ২য় ভগ্নাংশের হর `= a^3 - b^3`

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

    ৩য় ভগ্নাংশের হর `= a^2 + ab + b^2`

     :. ভগ্নাংশের হরগুলোর ল,সা,গু `= (a - b)^2 (a^2 + ab + b^2)`

      প্রাপ্ত ল,সা,গু কে প্রদত্ত ভগ্নাংশের হর দ্বারা ভাগ করলে ভাগফল হয় যথাক্রমে

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

     এখন, 

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

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

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

     `= ((a + b)^2 - 4ab)/(a^3 - b^3) = {((a + b)^2 - 4ab} xx (a - b))/((a^3 - b^3) xx (a - b))`

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

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

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

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

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

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

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

     নির্ণেয় সমহর বিশিষ্ট ভগ্নাংশগুলো হলো      `((a^4 - b^4) (a^2 + ab + b^2))/((a - b) (a^3 - b^3)), (a - b)^3/((a - b)(a^3 - b^3)), ((a - b)^2 (a + b))/((a - b) (a^3 - b^3))`

ক.  `D = (x^2 - y^2)/(x^3 - y^3)`

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

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

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

  খ. দেওয়া আছে, 

    `A = (x + 1)/(x - 1), B = (x^2 + x)/(x^2 + x - 2),`এবং ` C = x^2/(x^2 + 5x + 6)`

 `:. A -: B xx C = (x + 1)/(x - 1) -: (x^2 + x)/(x^2 + x - 2) xx x^2/(x^2 + 5x + 6)`

    `= (x + 1)/(x - 1) xx (x^2 + x - 2)/(x^2 + x) xx x^2/(x^2 + 5x + 6)`

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

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

    `= x/(x + 3)`

    `:. A -: B xx C = x/(x + 3)`  (প্রমাণিত)

  গ. `A = (x + 1)/(x - 1), B = (x^2 + x)/(x^2 + x - 2)`এবং `C = x^2/(x^2 + 5x + 6)`

      এখানে 

      A এর হর x = 1

      B এর হর `= x^2 + x - 2`

                 `= x^2 + 2x - x - 2`

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

   C এর হর `= x^2 + 5x + 6`

              `= x^2 + 3x + 2x + 6`

              `= x(x + 3) + 2(x + 3)`

              `= (x + 3) (x + 2)`

  :. A, B, C এর হরগুলোর ল,সা,গু = (x - 1) (x + 2) (x + 3)

   প্রাপ্ত ল,সা,গু কে প্রদত্ত ভগ্নাংশগুলির হর দ্বারা ভাগ করলে ভাগফল হয় যথাক্রমে

    (x + 2) (x + 3),  (x + 3) ও (x - 1)

  :. `A = (x + 1)/(x - 1) = ((x + 1)(x + 2)(x + 3))/((x - 1)(x + 2)(x + 3))`

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

  এবং `C = x^2/(x^2 + 5x + 6), = (x^2 xx (x - 1))/((x + 2)(x + 3) xx (x - 1)) = (x^2 - (x - 1))/((x - 1)(x + 2)(x + 3))`

  :. নির্ণেয় ভগ্নাংশগুলি `((x + 1)(x + 2)(x + 3))/((x - 1)(x + 2)(x + 3))`

   `((x^2 + x)(x + 3))/((x - 1)(x + 2)(x + 3)), (x^2 - (x - 1))/((x - 1)(x + 2)(x + 3))` (Ans)

ক. দেওয়া আছে, 

    ৩য় রাশির লব` = x^2 - 16`

  এবং ৩য় রাশির হর = `x^2 - 9`

   :. পার্থক্য `= (x^2 - 16)^2 - (x^2 - 9)^2`

   `(x^2)^2 - 2.x^2.16 + (16)^2 - {(x^2)^2 - 2.x^2.9 + 9^2}`

  `= x^4 - 32x^2 + 256 - x^4 + 18x^2 - 81`

  `= - 14x^2 + 175`

  `= 175 - 14x^2`

  খ. নির্ণেয় গুণফল `= (x^2 - 3x + 2)/(x^2 - 4x + 3) xx (x^2 - 5x + 6)/(x^2 - 7x + 12) xx (x^2 - 16)/(x^2 - 9)`

     `= (x^2 - 2x - x + 2)/(x^2 - 3x - x + 3) xx (x^2 - 3x - 2x + 6)/(x^2 - 4x - 3x + 12) xx (x^2 - 4^2)/(x^2 - 3^2)`

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

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

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

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

    উত্তর: `((x - 2)^2(x + 4))/((x - 3)^2 (x + 3))`


   গ. `(x^2 - 3x + 2)/(x^2 - 4x + 3) -: (x^2 - 5x + 6)/(x^2 - 7x + 12) xx (x^2 - 16)/(x^2 - 9)`

     `= (x^2 - 2x - x + 2)/(x^2 - 3x - x + 3) -: (x^2 - 3x - 2x + 6)/(x^2 - 4x - 3x + 12) xx (x^2 - 4^2)/(x^2 - 3^2)`

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

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

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

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