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2015-04-09T02:18:35+05:30

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P(x)  =  x^4  - 6 x^3 + 16 x^2 - 25 x + 10
         = (x^2)^2 - x^2 * 6 x + 16 * x^2  - 25 x + 10

Instead of long division, i am doing as follows:

     divisor:  x^2  - 2 x + k 
   so substitute  x^2 - 2x + k = 0    or  x^2 = 2 x - k  in  P(x)

 remainder  =    (2 x - k)^2  - 6 x ( 2x - k )  + 16 (2x - k) - 25 x + 10
               =  4x^2 - 4 k x + k^2 - 12 x^2 + 6 x k + 32 x - 16 k  - 25 x + 10
             =  - 8 x^2 + 2 k x + 7 x + k^2 - 16 k + 10
            =   - 8 (2 x - k) + 2  k x + 7 x + k^2 - 16 k + 10
           =  (- 9  + 2 k ) * x +  k^2 - 8 k + 10

   as the remainder is x + a  .  compare with the above expression:
           2 k - 9 = 1      and  so  k = 5
       10 - 8 k + k^2 = a      =>    a = - 5
=============================

long division

      x^2 - 2x + k  )  x^4 - 6 x^3 + 16 x^2 - 25 x + 10  (  x^2  - 4 x  + (8-k)
                           x^4  -2 x^3  + k x^2
                     ======================
                              - 4 x^3 + (16-k) x^2 - 25 x
                              - 4 x^3  + 8 x^2 - 4 k x
               ================================
                                       (8-k) x^2 + (4 k - 25) x  + 10
                                       8-k) x^2   - 2 (8-k) x   + k (8-k)
                    ===============================
                                    (2 k  - 9) x + 10 - 8 k + k^2

 we are given that    2 k - 9 = 1      hence,    k = 5
                  
           a = 10 - 8 k + k^2 = - 5 


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