Question:` a^4 - 27a^2 + 1` 

`a^4 - 27a^2 + 1 = (a^2)^2 - 2.a^2.1 + (1)^2 - 25a^2`

          ` = (a^2 - 1)^2 - (5a)^2`

           `= {(a^2 - 1) + 5a} {(a^2 - 1) - 5a}`

           `= (a^2 + 5a - 1) (a^2 - 5a - 1)`  (Ans)

Question:উৎপাদকে বিশ্লেষণ কর:

 `x^4 - 6x^2y^2 + y^4` 

`x^4 - 6x^2y^2 + y^4 = (x^2)^2 - 2x^2y^2 + (y^2)^2 - 4x^2y^2`

         `= (x^2 - y^2)^2 - (2xy)^2`

         `= (x^2 - y^2 + 2xy) (x^2 - y^2 - 2xy)`

         `= (x^2 + 2xy - y^2) (x^2 - 2xy - y^2)`

Question:উৎপাদকে বিশ্লেষণ কর:

 `(a^2 - b^2) (x^2 - y^2) + 4abxy` 

`a^2 - b^2) (x^2 - y^2) + 4abxy`

 `= a^2x^2 - a^2y^2 - b^2x^2 + b^2y^2 + 4abxy`
 
 `= a^2x^2 + 2axby + b^2y^2 - a^2y^2 + 2axby - b^2x^2`

 `= (a^2x^2 + 2abxy + b^2y^2) - (a^2y^2 - 2abxy + b^2x^2)`

 `= {(ax)^2 + 2ax.by + (by)^2} - {(ay)^2 - 2.ay.bx + (bx)^2}`

 `= (ax + by)^2 - (ay - bx)^2`

 `= {(ax + by) + (ay - bx)} {(ax + by) - (ay - bx)}`

 `= (ax + by + ay - bx)^2`

 `= {(ax + by) + (ay - bx)}{(ax + by) - (ay - bx)}`

 `= (ax + by + ay - bc) (ax + by - ay + bx)`  (Ans)

Question:উৎপাদকে বিশ্লেষণ কর:

 `4a^2 - 12ab + 9b^2 - 4c^2` 

`4a^2 - 12ab + 9b^2 -4c^2`

 `= (2a)^2 - 2.2a.3b + (3b)^2 - 4c^2`

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

 `= (2a - 3b + 2c) (2a - 3b - 2c)`   (Ans)

Question:উৎপাদকে বিশ্লেষণ কর:

 `9x^4 - 45a^2x^2 + 36a^4` 

`9x^4 - 45a^2x^2 + 36a^4`

 `= 9(x^4 - 5a^2x^2 + 4a^2)`

 `= 9(x^4 - 4a^2x^2 - a^2x^2 + 4a^4)`

 `= 9{x^2(x^2 - 4a^2) - a^2(x^2 - 4a^2)}`

 `= 9(x^2 - a^2) (x^2 - 4a^2)`

 `= 9(x^2 - a^2){x^2 - (2a)^2}`

 `= 9(x + a) (x - a) (x + 2a) (x - 2a)`  (Ans)

Question:উৎপাদকে বিশ্লেষণ কর:

    `a^2 + 6a + 8 - y^2 + 2y` 

`a^2 + 6a + 8 - y^2 + 2y`

      `= a^2 + 4a + 2a + 8 - y^2 + 2y`

     `= a(a + 4) +  2(a + 4) - y^2 + 2y`

     `= (a + 4) (a + 2) + 2y - y^2`

   ধরি, a + b = p

         a + 2 = q
         ------------
                2 = p - q  [বিয়োগ করে]

       :. p - q = 2

   এখন, `(a + 4) (a + 2) + 2y - y^2`

       `= pq + (p - q) y - y^2`

                  [:. a + 4 = p, a + 2 = q  এবং 2 = p - q]

       `= pq + py - qy - y^2`

       `= p(q + y) - y(q + y)`

       `= (q + y) (p - y)`

       `= (a + 2 + y) (a + 4 - y)`  [p, q]

       `= (a + y + 2) (a - y + 4)`  (Ans)


  বিকল্প সমাধান:   
      `a^2 + 6a + 8 - y^2 + 2y`

    `= a^2 + 6a + 9 - y^2 + 2y - 1`

    `= a^2 + 6a + 9 - (y^2 - 2y + 1)`

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

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

    `= (a + 3 + y - 1) (a + 3 - y + 1)`

    `= (a + y + 2) (a - y + 4)`  (Ans)

Question:উৎপাদকে বিশ্লেষণ কর:

       `16x^2 - 25y^2 - 8xz + 10yz` 

`16x^2 - 25y^2 - 8xz + 10yz`

      `= (4x)^2 - (5y)^2 - 2z(4x - 5y)`

      `= (4x + 5y) (4x - 5y) - 2z(4x - 5y)`

      `= (4x - 5y) (4x + 5y - 2z)`   (Ans)

Question:উৎপাদকে বিশ্লেষণ কর:

   `2b^2c^2 + 2c^2a^2 + 2a^2b^2 - a^4 - b^4 - c^4` 

`2b^2c^2 + 2c^2a^2 + 2a^2b^2 - a^4 - b^4 - c^4`

   `= 4a^2b^2 - a^4 - b^4 - c^4 - 2a^2b^2 + 2b^2c^2 + 2c^2a^2`

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

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

                                         `(-c^2) + (-c^2)a^2}`

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

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

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

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

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

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

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

Question:উৎপাদকে বিশ্লেষণ কর:

     `x^2 + 13x + 36` 

`x^2 + 13x + 36`

   `= x^2 + 9x + 4x + 36`

  `= x(x + 9) + 4(x + 9)`

 ` = (x + 9) (x + 4)`   (Ans)

Question:উৎপাদকে বিশ্লেষণ কর:

  `x^4 + x^2 - 20` 

`x^4 + x^2 - 20 = x^4 + 5x^2 - 4x^2 - 20`

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

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

  `= (x + 2) (x - 2) (x^2 + 5)`   (Ans)