Question:উৎপাদকে বিশ্লেষণ কর:

   `3(a^2 + 2a)^2 - 22(a^2 + 2a) + 40` 

`3(a^2 + 2a)^2 - 22(a^2 + 2a) + 40`

  `= a^2 + 2a = p`

  `= 3p^2 - 22p + 40`

  `= 3p^2 - 12p - 10p + 40`

  `= 3p(p - 4) - 10(p - 4)`

  `= (p - 4) (3p - 10)`

  `= (a^2 + 2a - 4) {3(a^2 + 2a) - 10}`  [p]

  `= (a^2 + 2a - 4) (3a^2 + 6a - 10)`    (Ans)

Question:উৎপাদকে বিশ্লেষণ কর:

  `14(x + z)^2 - 29(x + z) (x + 1) - 15(x + 1)^2` 

`14(x + z)^2 - 29(x + z) (x + 1) - 15(x + 1)^2`

   ধরি, `x + z = p`

   এবং `x + 1 = q`

  প্রদত্ত রাশি, 
     `= 14p^2 - 29pq - 15q^2`

     `= 14p^2 - 35pq + 6pq - 15q^2`

     `= 7p(2p - 5q) + 3q(2p - 5q)`

     `= (2p - 5q) (7p + 3q)`

     `= {2(x + z) - 5(x + 1)} {7(x + z) + 3(x + 1)}`

                                  [p, q এর মান বসিয়ে]

      `= (2x + 2z - 5x - 5) (7x + 7z + 3x + 3)`

      `= (2z - 3x - 5) (10x + 7z + 3)`  (Ans)

Question:উৎপাদকে বিশ্লেষণ কর:

  `(a - 1)x^2 + a^2 xy + (a + 1) y^2` 

`(a - 1)x^2 + a^2 xy + (a + 1) y^2`

    `a - 1 = p`

     `a + 1 = q`
   ------------------------------
     :.` (a - 1) (a + 1) = pq`   [গুণ করে]

         `a^2 - 1 = pq`

        `:. a^2 = pq + 1`

  প্রদত্ত রাশি  

 `= px^2 + (pq + 1) xy + qy^2`

 `= px^2 + pqxy + xy + qy^2`
 
`= px(x + qy) + y(x + qy)`

`= (x + qy) (px + y)`

`= {x + (a + 1)y} {(a - 1) x + y} `[p q এর মান বসিয়ে]

`= (x + ay + y) (ax - x + y)`   (Ans)

Question:উৎপাদকে বিশ্লেষণ কর:

  `24x^4 - 3x` 

`24x^2 - 3x`

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

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

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

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

Question:উৎপাদকে বিশ্লেষণ কর:

  `(a^2 + b^2)^3 + 8a^3b^3` 

`(a^2 + b^2)^3 + 8a^3b^3`

  ধরি, `a^2 + b^2 = p`

 :. প্রদত্ত রাশি

    `= p^3 + 8a^3b^3`

    `= p^3 + (2ab)^3`

    `= (p + 2ab) {p^2 - p.2ab + (2ab)^2}`

    `= (p + 2ab) (p^2 - 2abp + 4a^2b^2)`

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

                           [p এর মান বসিয়ে]

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

     `= (a + b)^2 (a^4 - 2a^3b + 6a^2b^2 - 2ab^3 + b^4)`   (Ans)

Question:উৎপাদকে বিশ্লেষণ কর:

  `x^3 + 3x^2 + 3x + 2` 

`x^3 + 3x^2 + 3x + 2`

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

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

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

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

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

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

Question:উৎপাদকে বিশ্লেষণ কর:

 `a^3 - 6a^2 + 12a - 9` 

`a^3 - 6a^2 + 12a - 9`

  `= a^3 - 6a^2 + 12a - 8 - 1`

  `= a^3 - 3.a^2.2 + 3.a.2^2 - 2^3 - 1`

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

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

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

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

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

Question:উৎপাদকে বিশ্লেষণ কর:

  `a^3 - 9b^3 + (a + b)^3` 

`a^3 - 9b^3 + (a + b)^3`

  `= a^3 - b^3 + (a + b)^3 - 8b^3`

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

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

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

  `= (a - b) (a^2 + ab + b^2) + (a - b) (a^2 + 4ab + 7b^2)`

  `= (a - b) (a^2 + ab + b^2 + a^2 + 4ab + 7b^2)`

  `= (a - b) (2a^2 + 5ab + 8b^2)`  (Ans)

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)