Question:উৎপাদকে বিশ্লেষণ কর:

  `8x^2 + 12x^2 + 6x - 63` 

`8x^2 + 12x^2 + 6x - 63`

  `= 8x^2 + 12x^2 + 6x + 1 - 64`

  `= (2x)^3 + 3.(2x)^2.1 + 3.2x. (1)^2 + (1)^3 - 64`

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

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

  `= (2x - 3) {(2x)^2 + 2.2x.1 + (1)^2 + 8x + 4 + 16}`

  `= (2x - 3) (4x^2 + 4x + 1 + 8x + 20)`

  `= (2x - 3) (4x^2 + 12x + 21)`  (Ans)

Question:উৎপাদকে বিশ্লেষণ কর:

  `8a^3 + b^3/(27)` 

`8a^3 + b^3/(27) = (2a)^3 + (b/3)^3`

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

  `= (2a + b/3) (4a^2 - (2ab)/3 + b^2/9)`

  `= ((6a + b)/3) ((36a^2 - 6ab + b^2)/9)`

  `= 1/(27) (6a + b) (36a^2 - 6ab + b^2)`  (Ans)

Question:উৎপাদকে বিশ্লেষণ কর:

 `a^3 - 1/8` 

`a^3 - 1/8 = a^3 - (1/2)^3`

  `= (a - 1/2) {a^2 + a.1/2 + (1/2)^2}`

  `= (a - 1/2) (a^2 + a/2 + 1/4)`

  `= ((2a - 1)/2) ((4a^2 + 2a + 1)/4)`

  `= 1/8 (2a - 1) (4a^2 + 2a + 1)`  (Ans)

Question:উৎপাদকে বিশ্লেষণ কর:

  `a^6/(27) - b^6` 

`a^6/(27) - b^6 = (a^2/3) - (b^2)^3`

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

   `= (a^2/3 - b^2) (a^4/9 + (a^2b^2)/3 + b^4)`  (Ans)

Question:উৎপাদকে বিশ্লেষণ কর:

 `4a^2 + 1/4a^2 - 2 + 4a - 1/a` 

`4a^2 + 1/4a^2 - 2 + 4a - 1/a`

  `= (2a)^2 + (1/(2a))^2 - 2.2a.1/(2a) + 2(2a - 1/(2a))`

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

  `= (2a - 1/(2a) (2a - 1/(2a) + 2)`  (Ans)

Question:উৎপাদকে বিশ্লেষণ কর:

  `(3a + 1)^3 - (2a - 3)^3` 

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

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

  `= (3a + 1 - 2a + 3) {(3a)^2 + 2.3a. 1 + (1)^2 + 3a. 2a + 1. 2a`

                            `- 3.3a - 3 + (2a)^2 - 2.2a .3 + (3)^2}`

   `= (a + 4) (9a^2 + 6a + 1 + 6a^2 + 2a - 9a - 3 + 4a^2 - 12a + 9`

   `= (a + 4) (19a^2 - 13a + 7)`   (Ans)

Question:উৎপাদকে বিশ্লেষণ কর:

  `(x + 5) (x - 9) - 15` 

`(x + 5) (x - 9) - 15`

  `= x^2 + 5x - 9x - 45 - 15`

  `= x^2 - 4x - 60`

  `= x^2 - 10x + 6x - 60`

  `= x(x - 10) + 6(x - 10)`

  `= (x + 6) (x - 10)`  (Ans)

Question:উৎপাদকে বিশ্লেষণ কর:

  `(x + 2) (x + 3) (x + 4) (x + 5) - 48` 

`(x + 2) (x + 3) (x + 4) (x + 5) - 48`

  `= (x + 2) (x + 5) (x + 3) (x + 4) - 48`

  `= (x^2 + 2x + 5x + 10) (x^2 + 3x + 4x + 12) - 48`

  `= (x^2 + 7x + 10) (x^2 + 7x + 12) - 48`

   ধরি, `x^2 + 7x + 10 = a`

  :.প্রদত্ত রাশি,

    `= x^2 + 7x + 12 = x^2 + 7x + 10 + 2 = a + 2`

    `= a(a + 2) - 48`

    `= a^2 + 2a - 48`

    `= a^2 + 8a - 6a - 48`

    `= a(a + 8) - 6(a + 8)`

    `= (a + 8) (a - 6)`

    `= (x^2 + 7x + 10 + 8) (x^2 + 7x + 10 - 6)`

                                        [a এর মান বসিয়ে]

    `= (x^2 + 7x + 18) (x^2 + 7x + 4)`

    `= (x^2 + 7x + 4) (x^2 + 7x + 18)`  (Ans)

Question:দেখাও যে, 
  `x^3 + 9x^2 + 26x + 24 = (x + 2) (x + 3) (x + 4)` 

বামপক্ষ  
  `= x^3 + 9x^2 + 26x + 24`

  `= x^3 + 9x^2 + 27x + 27 - x - 3`

  `= x^3 + 3.x^2.3 + 3.x.3^2 + 3^3 - (x + 3)`

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

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

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

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

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

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

   = ডানপক্ষ

    `:. x^2 + 9x^2 + 26x + 24 = (x + 2) (x + 3) (x + 4)` (দেখানো হলো)

Question:দেখাও যে,
  ` (x + 1) (x + 2) (3x - 1) (3x - 4) = (3x^2 + 2x - 1) (3x^2 + 2x - 8)` 

বামপক্ষ,  
 `= (x + 1) (x + 2) (3x - 1) (3x - 4)`

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

 `= (3x^2 + 3x - x - 1) (3x^2 + 6x - 4x - 8)`

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

    = ডানপক্ষ

   `:. (x + 1) (x + 2) (3x - 1) (3x - 4) = (3x^2 + 2x - 1) (3x^2 + 2x - 8)` (দেখানো হলো)