Let Q ( z ) = ( z − α 1 ) ⋯ ( z − α n ) be a polynomial of degree &gt; 1 with distinct roots outside the real line.We have ∑ j = 1 n 1 Q ′ ( α j ) = 0.Do we have a proof relying on rudimentary techniques?

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Let   Q  (  z  )  =  (  z  −      α    1    )  ⋯  (  z  −      α    n    ) be a polynomial of degree   &amp;gt;  1 with distinct roots outside the real line.We have      ∑          j      =      1        n        1                  Q        ′            (              α        j            )        =  0.Do we have a proof relying on rudimentary techniques?

Micah Hobbs · Accepted Answer

Let   n  ≥  1 and       α    1    ,  ⋯  ,      α    n    ∈      C   be distinct. If we put   Q  (  z  )  =  (  z  −      α    1    )  ⋯  (  z  −      α    n    ), then from the partial fraction decomposition, it follows that      1          Q      (      z      )        =      ∑          j      =      1              n            1                  Q        ′            (              α        j            )      (      z      −              α        j            )        .From this, we have      ∑          j      =      1              n            1                  Q        ′            (              α        j            )        =      lim                  |            z              |            →      ∞            ∑          j      =      1              n            z                  Q        ′            (              α        j            )      (      z      −              α        j            )        =      lim                  |            z              |            →      ∞            z          Q      (      z      )        =      {                            1          ,                          if           n          =          1                                      0          ,                          if           n          &amp;gt;          1                          .

Let Q(z)=(z−α_1)⋯(z−α_n) be a polynomial of degree >1 with distinct roots outside the real line. sum_(j=1)^n /(1)(Q'(alpha_j))=0. Do we have a proof relying on rudimentary techniques?

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