Determine the remainder when ( x 4 − 1 ) ( x 2 − 1 ) is divided by 1 + x + x 2

Question

Determine the remainder when   (      x    4    −  1  )  (      x    2    −  1  ) is divided by   1  +  x  +      x    2

komizmtk · Accepted Answer

The verbose equivalent is that                    (                  x          4                −        1        )        (                  x          2                −        1        )                            =        (                  x          4                −        x        )        (                  x          2                −        1        )        +        (        x        −        1        )        (                  x          2                −        1        )                                          =        (                  x          2                +        x        +        1        )        x        (        x        −        1        )        (                  x          2                −        1        )        +        (        x        −        1        )        (                  x          2                −        1        )            So   (  x  −  1  )  (      x    2    −  1  ) has the same remainder as   (      x    4    −  1  )  (      x    2    −  1  ) when dividing by       x    2    +  x  +  1.Answering the question in comments, yes, you can replace       x    2    +  x  +  1 with zero.Here, you can write:  (  x  −  1  )  (      x    2    −  1  )  =  (  x  −  1  )  (      x    2    +  x  +  1  )  −  (  x  −  1  )  (  x  +  2  )So you get the same remainder for   −  (  x  −  1  )  (  x  +  2  ) as   (  x  −  1  )  (      x    2    −  1  )  .

Gael Gardner · Accepted Answer

(      x    4    −  1  )  (      x    2    −  1  ) has a degree of six, we can write:  (      x    4    −  1  )  (      x    2    −  1  )  =  Q  (  x  )      [    1    +    x    +          x      2        ]    +  R  (  x  )Evaluate (1) at   {  ω  ,      ω    2    }                    R        (        ω        )                            =        (        ω        −        1        )        (                  ω          2                −        1        )        =        (        ω        −        1        )        (                  1          ω                −        1        )                            R        (                  ω          2                )                            =        (                  ω          2                −        1        )        (        ω        −        1        )        =        (        ω        −        1        )        (                  1          ω                −        1        )            This means that,  x  R  (  x  )  −  (  x  −  1  )  (  1  −  x  )  =  (  x  −  ω  )  (  x  −      ω    2    )Rearrange for the answer which is   R  (  x  )  =  3

Determine the remainder when ( x 4 </msup> − 1 ) ( x

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