মনে করি, `f(x) = 2x^4 - 3x^3 - 3x - 2`

  তাহলে, `f(2) = 2.2^4 - 3.2^4 - 3.2 - 2`

                  `= 32 - 24 - 6 - 2`

                    = 32 - 32

                    = 0

     :. (x - 2), f(x)  এর একটি উৎপাদক।

  এখন, `2x^4 - 3x^3 - 3x - 2`

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

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

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

        `= (x - 2) {x^2 (2x + 1) + 1(2x + 1)}`

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

ধরি, `f(x) = 4x^4 + 12x^3 + 7x^2 - 3x - 2`

  তাহলে, `f(- 1) = 4(- 1)^4 + 12 (- 1)^3 + 7 (- 1)^2 - 3(- 1) - 2`

    `= 4.1 + 12 (- 1) + 7.1 - 3 (- 1) - 2`

      = 4 - 12 + 7 + 3 - 2 

      = 14 - 14

      = 0

    :. {x - (- 1)} = (x + 1), f(x) এর একটি উৎপাদক।

   এখন,  `x^4 + 12x^3 + 7x^2 - 3x - 2`

         `= 4x^4 + 4x^3 + 8x^3 + 8x^2 - x^2 - x - 2x - 2`

         `= 4x^3 (x + 1) + 8x^2 (x + 1) - x(x + 1) - 2(x + 1)`

         `= (x + 1) (4x^3 + 8x^2 - x - 2)`

         `= (x + 1) {4x^2 (x + 2) - 1(x + 2)}`

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

         `= (x + 1) (x + 2) {(2x)^2 - 1^2}`

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

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

`x^6 - x^5 + x^4 - x^3 + x^2 - x`

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

 এখন, মনে করি, f(x) `= x^5 - x^4 + x^3 - x^2 + x - 1`

 :. f(1) `= (1)^5 - (1)^4 + (1)^3 - (1)^2 + (1) - 1`

         `= 1 - 1 + 1 - 1 + 1 - 1`

          = 3 - 3

         = 0

       :. (x - 1), f (x) এর একটি উৎপাদক।

  এখন, `x^5 - x^4 + x^3 - x^2 + x - 1`

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

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

   `= (x - 1) {(x^2)^2 + 2.x^2.1 + (1)^2 - x^2}`

   `= (x - 1) {(x^2 + 1)^2 - (x)^2}`

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

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

  `:. x^6 - x^5 + x^4 - x^3 + x^2 - x `

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

ধরি, `f(x) = 4x^3 - 5x^2 + 5x - 1`

 তাহলে, `f(1/4) = 4.(1/4)^3 - 5(1/4)^2 + 5 (1/4) - 1`

                 `= 4. 1/(64) - 5. 1/(16) + 5 . 1/4 - 1`

                 `= 1/(16) - 5/(16) + 5/4 - 1`

                 `= (1 - 5 + 20 - 16)/(16)`

                 `= (21 - 21)/(16)`

                 `= 0/(16)`

                   = 0

 `:. (x - 1/4)` অর্থাৎ (4x - 1), f(x) এর একটি উৎপাদক।

 এখন, `4x^3 - 5x^2 + 5x - 1`

  `= 4x^3 - x^2 - 4x^2 + x + 4x - 1`

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

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

ধরি, `f(x) = 18x^3 + 15x^2 - x - 2`

 তাহলে, `f(- 1/2) = 18 (- 1/2)^3 + 15(- 1/2)^2 - (- 1/2) - 2`

                    `= - 18 .1/2 + 15 . 1/4 + 1/2 - 2`

                    `= (- 9 + 15 + 2 - 8)/4`

                    `= (17 - 17)/4 = 0/4 = 0`

   `:. {x - (- 1/2)} = (x + 1/2)`

   অর্থাৎ (2x + 1), f(x) এর একটি উৎপাদক।

 এখন,  `18x^3 + 15x^2 - x - 2`

   `= 18x^3 + 9x^2 + 6x^2 + 3x - 4x - 2`

   `= 9x^2 (2x + 1) + 3x (2x + 1) - 2(2x + 1)`

   `= (2x + 1) (9x^2 + 3x - 2)`

   `= (2x + 1) (9x^2 + 6x - 3x - 2)`

   `= (2x + 1) {3x (3x + 2) - 1(3x + 2)}`

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