Find the remainder of f divided by g ( x ) = x 4 + x 2 + 1 if the remainder of f divided by h 1 ( x ) = x 2 + x + 1 is − x + 1 and the remainder of f divided by h 2 ( x ) = x 2 − x + 1 is 3 x + 5

Question

Find the remainder of f divided by   g  (  x  )  =      x    4    +      x    2    +  1 if the remainder of f divided by       h    1    (  x  )  =      x    2    +  x  +  1 is   −  x  +  1 and the remainder of f divided by       h    2    (  x  )  =      x    2    −  x  +  1 is   3  x  +  5

Jovan Wong · Accepted Answer

Let the polynomial be   P  (  x  ). It is given that for some polynomials   Q  (  x  )  ,      Q    1    (  x  )  P  (  x  )  =  (      x    2    +  x  +  1  )  Q  (  x  )  +  1  −  x  P  (  x  )  =  (      x    2    −  x  +  1  )      Q    1    (  x  )  +  3  x  +  5Now it is well known that       x    2    +  x  +  1 has the zeros as   ω  ,      ω    2   So We get  P  (  ω  )  =  1  −  ω  P  (      ω    2    )  =  1  −      ω    2    P  (  −  ω  )  =  3  ω  +  5  P  (  −      ω    2    )  =  −  3      ω    2    +  5Now let us assume for some polynomial       Q    2    (  x  ) we have  P  (  x  )  =  (      x    4    +      x    2    +  1  )      Q    2    (  x  )  +  A      x    3    +  B      x    2    +  C  x  +  DUsing the fact that       x    4    +      x    2    +  1 has the zeros   ω  ,      ω    2    ,  −  ω  ,  −      ω    2   we get four linear equations as:      [                            1                                      ω                          2                                                ω                          1                                      1                          ω                                      ω                          2                                                1                                      −          1                                      ω                          2                                                −          ω                          1                                      −          1                          ω                          −                      ω                          2                                                1                      ]        [                            A                                      B                                      C                                      D                      ]    =      [                            1          −          ω                                      1          −                      ω                          2                                                            5          −          3          ω                                      5          −          3                      ω                          2                                            ]  Now let us solve by Cramer&#039;s rule. The determinant of the matrix is   Δ  =  12,       Δ    1    =  −  24,       Δ    2    =  24,       Δ    3    =  12 and       Δ    4    =  60. Thus the values of A,B,C,D are                          A        =                              Δ                          1                                Δ                =        −        2                                  B        =                              Δ                          2                                Δ                =        2                                  C        =                              Δ                          3                                Δ                =        1                                  D        =                              Δ                          4                                Δ                =        5            So the required remainder is  −  2      x    3    +  2      x    2    +  x  +  5

Find the remainder of f divided by g ( x ) = x 4 </msup> + x 2

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