Question:সরল কর:

 ক. `(4a - 3b)^3 - 3(4a - 3b)^2 (2a - 3b) + 3(4a - 3b)`

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

 খ. `(2x + y)^3 + 3(2x + y)^2 (2x - y) + 3(2x + y) (2x - y)^2 + (2x - y)^3`

 গ. `(7x + 3b)^3 - (5x + 3b)^3 - 6x(7x + 3b) (5x + 3b)`

 ঘ. `(x - 15)^3 + (16 - x)^3 + 3(x  - 15) (16 - x)`

 ঙ. `(a + b + c)^3 - (a - b - c)^3 - 6(b + c) {a^2 - (b + c)^2}`

 চ. `(m + n)^6 - (m - n)^6 - 12mn(m^2 - n^2)^2`

 ছ. `(x + y) (x^2 - xy + y^2) + (y + z) (y^2 - yz + z^2) + (z + x)`

     ` (z^2 - zx + x^2)`

 জ. `(2x + 3y - 4z)^3 + (2x - 3y + 4z)^3 + 12x{4x^2 - (3y - 4z)^2}` 

Answer ক. `(4a - 3b)^3 - 3(4a - 3b)^2 (2a - 3b) + 3(4a - 3b)`

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


   মনে করি, 4a - 3b = x  এবং 2a - 3b = y

 :. প্রদত্ত রাশি  

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

  `= (x - y)^3`

  `= {(4a - 3b) - (2a - 3b)}^3`

                  [x এবং y এর মান বসিয়ে]

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

  `= (2a)^3`

  `= 8a^3`

  খ. `(2x + y)^3 + 3(2x + y)^2 (2x - y) + 3(2x + y) (2x - y)^2 +`

                                                      ` (2x - y)^3`

  ধরি, 2x + y = p এবং 2x - y = q

  :. প্রদত্ত রাশি `= p^3 + 3p^2q + 3pq^2 + q^3`

                 `= (p + q)^3`

                 `= (2x + y + 2x - y)^3`  [মান বসিয়ে]

                 `= (4x)^3`

                 `= 64x^3`

  গ.`(7x + 3b)^3 - (5x + 3b)^3 - 6x(7x + 3b) (5x + 3b)`

      ধরি, 7x + 3b = p এবং 5x + 3b = q

    :. p - q = 7x + 3b - 5x - 3b

    :. p - q = 2x

   :. প্রদত্ত রাশি 

  `= p^3 - q^3 - 6x.pq`

  `= p^3 - q^3 - 3.pq. 2x`

  `= p^3 - q^3 - 3pq (p - q);` [:. 2x = p - q]

  `= (p - q)^3`

  `= (2x)^3`

  `= 8x^3`    (Ans)

 ঘ.`(x - 15)^3 + (16 - x)^3 + 3(x - 15) (16 - x)`

    ধরি, x - 15 = p এবং 16 - x = q

   :. p + q = x - 15 + 16 - x

   :. p + q = 1

   প্রদত্ত রাশি 
      `= p^3 + q^3 + 3pq`

      `= p^3 + q^3 + 3pq.1`

      `= p^3 + q^3 + 3pq (p + q)`   [:. p + q = 1]

      `= (p + q)^3`

      `= 1^3`

        = 1


  ঙ.`(a + b + c)^3 - (a - b - c)^3 - 6(b + c) {a^2 - (b + c)^2}`

  `= (a + b + c)^3 - (a - b - c)^3 - 3 xx 2(b + c) {a^2 - (b + c)^2}`

    মনে করি, a + b + c = x a - b - c = y

    তাহলে x - y `= (a + b + c) - (a - b - c)`

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

            `= 2b + 2c`

            `= 2(b + c)`

  এবং xy = (a + b + c) (a - b - c)

           = { a + (b + c)} {a - (b + c)}

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

 :. প্রদত্ত রাশি, `= x^3 - y^3 - 3(x - y)xy` [মান বসিয়ে]

                 `= (x - y)^3`

                 `= {2(b + c)}^3`  [(x - y) এর মান বসিয়ে]

                 `= 8(b + c)^3`  (Ans)


  চ.`(m + n)^6 - (m - n)^6 - 12mn (m^2 - n^2)^2`

   `= (m + n)^6 - (m - n)^6 - 3(m^2 - n^2)^2 xx 4mn`

  `= {(m + n)^2}^3 - {(m - n)^2}^3 - 3(m + n)^2`

                          `(m - n)^2 {(m + n)^2 - (m - n)^2}`

 `= {(m + n)^2 - (m - n)^2}^3`

  `= (4mn)^3 = 64m^3n^3`

 ছ. `(x + y)(x^2 - xy + y^2) + (y + z) (y^2 - yz + z^2)`

                                  `+ (z + x) (z^2 - zx + x^2)`

  `= (x^3 + y^3) + (y^3 + z^3) + (z^3 + x^3)`

  `= x^3 + y^3 + y^3 + z^3 + z^3 + x^3`

  `= 2x^3 + 2y^3 + 2z^3`

  `= 2(x^3 + y^3 + z^3)`

 জ.  `(2x + 3y - 4z)^3 + (2x - 3y + 4z)^3 + 12x{(2x) - (3y - 4z)^2}`

  `= (2x + 3y - 4z)^3 + (2x - 3y + 4z)^3 + 3.4x(2x + 3y - 4z) (2x - 3y + 4z)`

       2x + 3y - 4z = p

      2x - 3y + 4z = q

      4x = p + q;

    `= p^3 + q^3 + 3(p + q).pq`

    `= (p + q)^3`

    `= (2x + 3y - 4z + 2x - 3y + 4z)^3`  [মান বসিয়ে]

    `= (4x)^3`

    `= 64x^3` 

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