ক. ‍১ম রাশির হর `= 2x^2 + 7x - 4`

   এবং ২য় রাশির লব` = 3x^2 + 11x - 4`

   :. নির্ণেয় ভাগফল` = (2x^2 + 7x - 4)/(3x^2 + 11x - 4)`

                      `= (2x^2 + 8x - x - 4)/(3x^2 + 12x - x - 4)`

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

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

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

                       উত্তর: `(2x - 1)/(3x - 1)`

  খ. `(2x^2 - 7x + 3)/(2x^2 + 7x - 4) xx (3x^2 + 11x - 4)/(3x^2 + 8x - 3) -: (2x^2 + x - 15)/(2x^2 - 11x + 15)`

 `= (2x^2 - 6x - x + 6)/(2x^2 + 8x - x - 4) xx (3x^2 + 12x - x - 4)/(3x^2 + 9x - x - 3) -: (2x^2 + 6x - 5x - 15)/(2x^2 - 6x - 5x + 15)`

 `= (2x (x - 3) - 1(x - 3))/(2x(x + 4) - 1(x + 4)) xx (3x(x + 4) - 1(x + 4))/(3x(x + 9) - 1(x + 3)) -: (2x(x + 3) - 5(x + 3))/(2x (x - 3) - 5(x - 3))`

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

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

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

    উত্তর: `((x - 3))^2/((x + 3))^2`

 গ. `(2x^2 - 7x + 3)/(2x^2 + 7x - 4), (3x^2 + 11x - 4)/(3x^2 + 8x - 3), (2x^2 + x - 15)/(2x^2 - 11x + 15)`

  এখানে,

  ১ম ভগ্নাংশের হর `= 2x^2 + 7x - 4`

                   `= 2x^2 + 8x - x - 4`

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

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

  ২য় ভগ্নাংশের হর` = 3x^2 + 8x - 3`

                   ` = 3x^2 + 9x - x - 3`

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

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

  ৩য় ভগ্নাংশের হর `= 2x^2 - 11x + 15`

                    `= 2x^2 - 6x - 5x + 15`

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

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

     :. ভগ্নাংশগুলির হরগুলোর ল,সা,গু

        ` =  (x -: 4) (2x - 1) (x + 3)(3x - 1)(x - 3)(2x - 5))`

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

            `(x + 3) (3x - 1)(x - 3)`

  `(2x - 5), (x + 4) (2x - 1) (x - 3) (2x - 5) (x + 4) (2x - 1) (x + 3) (3x - 1)`

 এখন, `(2x^2 - 7x + 3)/(2x^2 + 7x - 4) = ((2x^2 - 7x + 3) xx (x + 3)(3x - 1)(x - 3)(2x - 5))/((2x^2 + 7x - 4) xx (x + 3) (3x - 1) (x - 3)(2x - 5))`

 `= ((2x^2 - 7x + 3)(x^2 - 3^2)(6x^2 - 15x - 2x + 5))/((x + 4)(2x - 1)(x + 3)(3x - 1)(x - 3)(2x - 5))`

 `= ((x^2 - 9)(2x^2 - 7x + 3)(6x^2 - 17x + 5))/((3x - 1)(2x - 5)(2x - 1)(x - 3)(x + 3)(x + 4))`

 `(3x^2 + 11x - 4)/(3x^2 + 8x - 3) = ((3x^2 + 11x - 4) xx (x + 4)(2x - 1)(x - 3)(2x - 5))/((3x^2 + 8x - 3) xx (x + 4)(2x - 1)(x - 3)(2x - 5))`

 `= ((3x^2 + 11x - 4)(2x^2 - x + 8x - 4)(2x^2 - 5x - 6x + 15))/((x + 3)(3x - 1)(x + 4)(2x - 1)(x - 3)(2x - 5))`

 `= ((3x^2 + 11x - 4)(2x^2 + 7x - 4)(2x^2 - 11x + 15))/((3x - 1)(2x - 5)(2x - 1)(x - 3)(x + 3)(x + 4))`

 এবং `(2x^2 + x - 15)/(2x^2 - 11x + 15) = ((2x^2 + x - 15) xx (x + 4)(2x - 1)(x + 3)(3x - 1))/((2x^2 - 11x + 15) xx (x + 4)(2x - 1)(x + 3)(3x - 1))`

 `= ((2x^2 + x - 15)(2x^2 - x + 8x - 4)(3x^2 - x + 9x - 3))/((x - 3)(2x - 5)(x + 4)(2x - 1)(x + 3)(3x - 1))`

 `= ((2x^2 + x - 15)(2x^2 + 7x - 4)(3x^2 + 8x - 3))/((3x - 1)(2x - 5)(2x - 1)(x - 3)(x + 3)(x + 4))`

 :.  নির্ণেয় ভগ্নাংশগুলো` = ((x^2 - 9)(2x^2 - 7x + 3)(6x^2 - 17x + 5))/((3x - 1)(2x - 5)(2x - 1)(x - 3)(x + 3)(x + 4))`

 `= ((3x^2 + 11x - 4)(2x^2 + 7x - 4)(2x^2 - 11x + 15))/((3x - 1)(2x - 5)(2x - 1)(x - 3)(x + 3)(x + 4))`

 `= ((2x^2 + x - 15)(2x^2 + 7x - 4)(3x^2 + 8x - 3))/((3x - 1)(2x - 5)(2x - 1)(x - 3)(x + 3)(x + 4))`

ক. ৩য় রাশি 

 `= (m^3 + n^3)/(m^3 - m^2 n + mn^2)`

 `= ((m + n)(m^2 - mn + n^2))/(m(m^2 - mn + n^2))`

 `= (m + n)/m`

  উত্তর:  `(m + n)/m`

 খ. নির্ণেয় গুণফল 

 ` = (1 - z/(x + y)) (x/(x + y + z) + x/(x + y - z))`

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

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

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

 `= (2x(x + y))/((x + y)(x + y + z))`

 `= (2x)/(x + y + z)`

  উত্তর:  `(2x)/(x + y + z)`

 
  গ. `(mn)/(m - n), (m^3 + n^3)/(m^3 - m^2 n + mn^2), (m^3 - n^3)/(m^2 + mn + n^2)`

   এখানে, ১ম ভগ্নাংশের হর = m - n

   ২য় ভগ্নাংশের হর = `m^3 - m^2 n + mn^2`

                     =` m(m^2 - mn + n^2)`

  ৩য় ভগ্নাংশের হর =` m^2 + mn + n^2`

  :. হরগুলোর ল,সা,গু `= m(m - n) (m^2 + mn + n^2) (m^2 - mn + n^2)`

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

  `m(m^2 + mn + n^2) (m^2 - mn + n^2), (m - n)(m^2 + mn + n^2) m(m - n) (m^2 - mn + n^2)`

  এখন, 

 :. `(mn)/(m - n) = (mn xx m(m^2 + mn + n^2)(m^2 - mn + n^2))/((m - n) xx m(m^2 + mn + n^2) (m^2 - mn + n^2))`

  `= (m^2 n(m^2 - mn + n^2) (m^2 + mn + n^2))/(m(m^2 - mn + n^2)(m^3 - n^3))`

 `(m^3 + n^3)/(m^3 - m^2 n + mn^3) = ((m^3 + n^3) xx (m - n) (m^2 + mn + n^2))/((m^3 - m^2 n + mn^2) xx (m - n)(m^2 + mn + n^2))`

   `= ((m^3 - n^3)(m^3 + n^3))/(m(m^2 - mn + n^2) (m^3 - n^3))`

  এবং `(m^3 - n^3)/(m^2 + mn + n^2) = ((m^3 - n^3) xx m(m - n) (m^2 - mn + n^2))/((m^2 + mn + n^2) xx m(m - n) (m^2 - mn + n^2))`

  `= (m(m - n)(m^2 - mn + n^2)(m^3 - n^3))/(m(m^2 - mn + n^2)(m^3 - n^3))`

  নির্ণেয় ভগ্নাংশগুলো

 ` (m^2 n(m^2 - mn + n^2)(m^2 + mn + n^2))/(m(m^2 - mn + n^2) (m^3 - n^3))`

 `= ((m^3 - n^3)(m^3 + n^3))/(m(m^2 - mn + n^2)(m^3 - n^3)), (m(m - n)(m^2 - mn + n^2)(m^3 - n^3))/(m(m^2 - mn + n^2)(m^3 - n^3))`

coming soon