Let the roots of the polynomial P ( x ) = x 3 − x + 1 be a, b, c. Construct the polynomial with roots a 4 , b 4 , c 4 and degree 3

Question

Let the roots of the polynomial   P  (  x  )  =      x    3    −  x  +  1 be a, b, c. Construct the polynomial with roots       a    4    ,      b    4    ,      c    4   and degree 3

tomatoland45wt8wm · Accepted Answer

You are on the right track, if you write       u                  1        4              =  x and form an equation in u which can be changed into the form of a polynomial.So,      u                  3        4              −      u                  1        4              +  1  =  0    ⟹        u                  1        4              (      u                  1        2              −  1  )  =  −  1    ⟹        u                  1        2              −  1  =  −      1          u                        1          4                    Now squaring both sides,  u  −  2      u                  1        2              +  1  =      1          u                        1          2                        ⟹    u  +  1  =      1          u                        1          2                      +  2      u                  1        2            Squaring again,      u    2    +  2  u  +  1  =      1    u    +  4  +  4  uSo the polynomial is      u    3    −  2      u    2    −  3  u  −  1  =  0

Derick Richard · Accepted Answer

Maybe I was wrong in calculations, one needs to check.Vieta:   a  +  b  +  c  =  0,   a  b  +  a  c  +  b  c  =  −  1,   a  b  c  =  −  1Let&#039;s do some algebraic manipulations to obtain the same for       a    4  ,       b    4  ,       c    4    (  a  +  b  +  c      )    2    =      a    2    +      b    2    +      c    2    +  2  (  a  b  +  b  c  +  a  c  )  ⇒      a    2    +      b    2    +      c    2    =  2  (  a  b  +  a  c  +  b  c      )    2    =      a    2        b    2    +      a    2        c    2    +      b    2        c    2    +  2  a  b  c  (  a  +  b  +  c  )  ⇒      a    2        b    2    +      a    2        c    2    +      b    2        c    2    =  1  (      a    2    +      b    2    +      c    2        )    2    =      a    4    +      b    4    +      c    4    +  2  (      a    2        b    2    +      a    2        c    2    +      b    2        c    2    )  ⇒      a    4    +      b    4    +      c    4    =  2      a    2        b    2        c    2    =  (  a  b  c      )    2    =  1  (      a    2        b    2    +      a    2        c    2    +      b    2        c    2        )    2    =      a    4        b    4    +      a    4        c    4    +      b    4        c    4    +  2      a    2        b    2        c    2    (      a    2    +      b    2    +      c    2    )  ⇒      a    4        b    4    +      a    4        c    4    +      b    4        c    4    =  −  3It&#039;s ok, because there are complex roots in   a  ,  b  ,  c      a    4        b    4        c    4    =  (  a  b  c      )    4    =  1Reverse Vieta:       x    3    −  2      x    2    +  3  x  −  1 is polynomial with roots       a    4  ,       b    4  ,       c    4

Let the roots of the polynomial P ( x ) = x 3 </msup> − x + 1

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