Question

if the polynomial x^4-6x^3-16x^2+25x+10 is divided by another polynomial x^2-2x+k , the remainder comes out to be x+a, find k and a.

It is not +16x^2. You may think like that. Answer the question by keeping it as -16x^2 only.

Answer 1

x^2-2x+k )     x^4  - 6 x^3  - 16 x^2  + 25  x  + 10 ( x^2 - 4 x  - (24+k)
                    x^4  - 2x^3 + k x^2
                   ===================
                          - 4 x^3  -(16+k) x^2  + 25 x
                          - 4 x^3  + 8 x^2  - 4 k x
                         =====================
                                     - (16 + k + 8) x^2  + (25+4k) x   + 10
                                     -(24 +k) x^2 + 2(24+k) x - (24+k) k
                                    ============================
                                                 (-23 + 2 k) x + (k^2+ 24 k +10)
 -23 + 2 k = 1       =>  k = 12
 k^2 + 24 k + 10 =  a      =>    a = 442
  reminder = x + 442


verification
 ========================================
 x^2 - 2x + 12  )  x^4  - 6 x^3  - 16 x^2 + 25 x + 10 (  x^2  - 4 x - 36
                              x^4 - 2 x^3  + 12 x^2
                             ====================
                                     - 4 x^3 - 28 x^2  + 25 x
                                     - 4 x^3  + 8 x^2  - 48 x
                                         ===============
                                              - 36 x^2  + 73 x  + 10
                                             - 36 x^2 + 72 x   - 432
                                     =========================
                                                                1 x  +  442

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