What is the remainder on division of z 400 + z 303 + 1 by z 4 − 1?I am asked to determine the remainder when the polynomial f ( z ) = z 400 + z 303 + 1 is divided by the polynomial g ( z ) = z 4 − 1.I expressed f as f ( z ) = h ( z ) ( z 4 − 1 ) + r ( z ) where r(z) is a polynomialRealising that z 4 − 1 = ( z 2 − 1 ) ( z 2 + 1 ) = ( z − 1 ) ( z + 1 ) ( z 2 + 1 ), I expressed f as f ( z ) = h ( z ) ( z − 1 ) ( z + 1 ) ( z 2 + 1 ) + r ( z )I also wrote f as f ( z ) = q ( z ) ( z − 1 ) + 3, since f ( 1 ) = 3, and q is some polynomial.So I had f ( z ) = q ( z ) ( z − 1 ) + 3 = h ( z ) ( z − 1 ) ( z + 1 ) ( z 2 + 1 ) + r ( z )Hence, for some constant k, q ( z ) = h ( z ) ( z + 1 ) ( z 2 + 1 ) + kWe know f ( − 1 ) = 1From f ( z ) = q ( z ) ( z − 1 ) + 3, we have f ( − 1 ) = q ( − 1 ) ( − 2 ) + 3 = 1, which implies q ( − 1 ) = 1.Then q ( − 1 ) = h ( − 1 ) ( 0 ) + k = 1. Thus, k = 1We then have q ( z ) = h ( z ) ( z + 1 ) ( z 2 + 1 ) + 1, and from this, we have f ( z ) = q ( z ) ( z − 1 ) + 3 = ( h ( z ) ( z + 1 ) ( z 2 + 1 ) + 1 ) ( z − 1 ) + 3. Which simplifies to f ( z ) = h ( z ) ( z 4 − 1 ) + z + 2And gives r ( z ) = z + 2.But I am given the answer is r ( z ) = z 3 + 2. What is going wrong here? Can I use modular arithmetic instead? How?

Question

What is the remainder on division of       z          400        +      z          303        +  1 by       z    4    −  1?I am asked to determine the remainder when the polynomial   f  (  z  )  =      z          400        +      z          303        +  1 is divided by the polynomial   g  (  z  )  =      z    4    −  1.I expressed f as   f  (  z  )  =  h  (  z  )  (      z    4    −  1  )  +  r  (  z  ) where r(z) is a polynomialRealising that       z    4    −  1  =  (      z    2    −  1  )  (      z    2    +  1  )  =  (  z  −  1  )  (  z  +  1  )  (      z    2    +  1  ), I expressed f as  f  (  z  )  =  h  (  z  )  (  z  −  1  )  (  z  +  1  )  (      z    2    +  1  )  +  r  (  z  )I also wrote f as   f  (  z  )  =  q  (  z  )  (  z  −  1  )  +  3, since   f  (  1  )  =  3, and q is some polynomial.So I had   f  (  z  )  =  q  (  z  )  (  z  −  1  )  +  3  =  h  (  z  )  (  z  −  1  )  (  z  +  1  )  (      z    2    +  1  )  +  r  (  z  )Hence, for some constant k,   q  (  z  )  =  h  (  z  )  (  z  +  1  )  (      z    2    +  1  )  +  kWe know   f  (  −  1  )  =  1From   f  (  z  )  =  q  (  z  )  (  z  −  1  )  +  3, we have   f  (  −  1  )  =  q  (  −  1  )  (  −  2  )  +  3  =  1, which implies   q  (  −  1  )  =  1.Then   q  (  −  1  )  =  h  (  −  1  )  (  0  )  +  k  =  1. Thus,   k  =  1We then have   q  (  z  )  =  h  (  z  )  (  z  +  1  )  (      z    2    +  1  )  +  1, and from this, we have   f  (  z  )  =  q  (  z  )  (  z  −  1  )  +  3  =  (  h  (  z  )  (  z  +  1  )  (      z    2    +  1  )  +  1  )  (  z  −  1  )  +  3. Which simplifies to   f  (  z  )  =  h  (  z  )  (      z    4    −  1  )  +  z  +  2And gives   r  (  z  )  =  z  +  2.But I am given the answer is   r  (  z  )  =      z    3    +  2. What is going wrong here? Can I use modular arithmetic instead? How?

Maffei2el · Accepted Answer

Step 1Since                              z                      400                          +                  z                      303                          +        1                            =                  z                      400                          −        1        +        (                  z                      300                          −        1        )                  z          3                +                  z          3                +        2                                          =        (                  z          4                −        1        )        (                  z                      396                          +                  z                      392                          +        …        +        1        )        +        (                  z          4                −        1        )        (                  z                      296                          +                  z                      292                          +        …        +        1        )                  z          3                +                  z          3                +        2            Step 2It follows that       z    3    +  2 is the asked remainder.

Noe Cowan · Accepted Answer

Step 1Note that we can write   f  (  z  )  =  h  (  z  )  (      z    4    −  1  )  +  r  (  z  ) where r(z) is a polynomial of degree at most three (one less than the degree of       z    4    −  1).An unknown polynomial of degree 3 has four coefficients to be determined. If we successively use   z  =  1  ,  z  =  −  1  ,  z  =  i  ,  z  =  −  i - the four roots of       z    4    −  1 the term involving h(z) will vanish.This gives us   f  (  1  )  =  r  (  1  )  =  3,   f  (  −  1  )  =  r  (  −  1  )  =  1,   f  (  i  )  =  r  (  i  )  =  2  −  i,   f  (  −  i  )  =  r  (  −  i  )  =  2  +  iIf   r  (  z  )  =  a      z    3    +  b      z    2    +  c  z  +  d we have  a  +  b  +  c  +  d  =  3  −  a  +  b  −  c  +  d  =  1From which we obtain   b  +  d  =  2 and   a  +  c  =  1Also  −  a  i  −  b  +  c  i  +  d  =  2  −  iFrom which −b+d=2 so b=0,d=2 and −a+c=−1 so that a=1,c=0r(z)=z3+2Step 2Where you seem to have gone wrong is when you get to  f  (  z  )  =  q  (  z  )  (  z  −  1  )  +  3  =  h  (  z  )  (  z  −  1  )  (  z  +  1  )  (      z    2    +  1  )  +  r  (  z  )From this you can conclude that   (  z  −  1  )      (    q    (    z    )    −    h    (    z    )    (    z    +    1    )    (          z      2        +    1    )    )    =  r  (  z  )  −  3 but that does not mean that the term in large brackets is a constant. It is that assumption which puts you off track.

What is the remainder on division of z^{400}+z^{303}+1 by z^4-1?

Answered question

Answer & Explanation

New Questions in Algebra II