If α , β , γ are the roots of the equation x 3 + x 2 − x + 1 = 0, find the value of ∏ ( 1 α 3 + 1 β 3 − 1 γ 3 )

Question

If   α  ,  β  ,  γ are the roots of the equation       x    3    +      x    2    −  x  +  1  =  0, find the value of   ∏      (          1              α        3              +          1              β        3              −          1              γ        3              )

Morisio64moc · Accepted Answer

1. Let   P  (  x  )  =      x    3    +      x    2    −  x  +  1. Since all the zeros of   P  (  x  ) are non-zero,   gcd  (  P  (  x  )  ,      x    3    )  =  1. Then the Bézout&#039;s identity tells that we can find polynomials   A  (  x  ) and   B  (  x  ) satisfying  A  (  x  )  P  (  x  )  +  B  (  x  )      x    3    =  1.Although there is a systematic way of determining   A  (  x  ) and   B  (  x  ), called the extended GCD algorithm, it is not hard to see that   A  (  x  )  =  x  +  1 and   B  (  x  )  =  −  x  −  2 from the computation  (  x  +  1  )  (      x    2    −  x  +  1  )  =      x    3    +  1      ⟹      (  x  +  1  )  P  (  x  )  =      x    4    +  2      x    3    +  1.The upshot of this computation is that, if   x  =      x    0   is any zero of   P  (  x  )  =  0, then  B  (      x    0    )      x    0    3    =  1    and hence        1          x      0      3        =  B  (      x    0    )  =  −      x    0    −  2.2. Plugging this to OP&#039;s product, we get      ∏          cyc            (          1              α        3              +          1              β        3              −          1              γ        3              )    =      ∏          cyc            (    −    α    −    β    +    γ    −    2    )    =      ∏          cyc            (    2    γ    −    1    )    =  (  −  2      )    3    P      (          1      2        )    =  −  7.

If α , β , γ are the roots of the equ

Answered question

Answer & Explanation

New Questions in Precalculus