ধরি, `f(x) = 2x^2 - x - 3`

 তাহলে, `f(- 1) = 2 (-1)^2 - (-1) - 3`

             = 2 + 1 - 3

             = 0

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

 এখন, `2x^2 - x - 3`

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

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

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

ধরি, `f(x) = 3x^2 - 7x - 6`

 তাহলে, `f(3) = 3 (3)^2 - 7.3 - 6`

   `= (3 xx 9) - 21 - 6`

    = 27 - 27

    = 0

  `:. (x - 3), f(x)` এর একটি উৎপাদক।

  এখন, `3x^2 - 7x - 6`

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

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

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

ধরি, `f(x) = x^3 + 2x^2 - 5x - 6`

 তাহলে, `f(2) = (2)^3 + 2(2)^2 - 5(2) - 6`

         = 8 + 2.4 - 10 - 6

         = 8 + 8 - 10 - 6

         = 0

 `:. (x - 2), f(x)`  এর একটি উৎপাদক।

 এখন, `x^3 + 2x^2 - 5x - 6`

  `= x^3 - 2x^2 + 4x^2 - 8x + 3x - 6`

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

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

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

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

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

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

ধরি, `f(x) = x^3 + 4x^2 + x - 6`

তাহলে, `f(1) = (1)^3 + 4(1)^2 + 1 - 6`

       `= 1 + 4.1 + 1 - 6`

         = 6 - 6

         = 0

   :. (x - 1), f(x)  এর একটি উৎপাদক।

  এখন, `x^3 + 4x^2 + x - 6`

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

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

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

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

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

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

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

মনে করি,  `f(a) = a^4 - 4a + 3`

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

      `= 1 - 4 + 3 = 4 - 4 = 0`

   `:. (a - 1), f(a)` এর একটি উৎপাদক।

  এখন, `a^4 - 4a + 3`

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

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

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

 আবার মনে করি, `g(a) = a^3 + a^2 + a - 3`

 তাহলে, `g(1) = (1)^3 + (1)^2 + (1) - 3`

               `= 1 + 1 + 1 - 3`

               `= 3 - 3 = 0`

  `:. (a - 1), g(a)` এর একটি উৎপাদক।

 এখন, `a^3 + a^2 + a - 3`

  `= a^3 - a^2 + 2a^2 - 2a + 3a - 3`

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

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

 `:. a^4 - 4a + 3 = (a - 1) (a - 1) (a^2 + 2a + 3)`  (Ans)