Let P be a polynomial of degree n , n ≥ 2, P ( X ) = ∏ i = 1 n ( X − z i ) m i with zeros z 1 , z 2 , … , z n Decompose the rational fraction P ′ P My thoughts: P ( X ) = ∏ i = 1 n ( X − z i ) m i P ′ ( X ) = ∑ i = 1 n m i ( X − z i ) m i − 1 ∏ ? ? ( X − z i ) m ? I don't know with what i should fill the ?Then: P ′ P = ∑ i = 1 n m i ( X − z i ) m i − 1 ∏ ? ? ( X − z i ) m i ∏ i = 1 n ( X − z i ) m i = ? I'm stuck here

Question

Let   P be a polynomial of degree   n ,   n  ≥  2,   P  (  X  )  =      ∏          i      =      1        n              (      X      −              z        i            )                      m        i            with zeros       z    1    ,      z    2    ,  …  ,      z    n  Decompose the rational fraction                     P        ′            P      My thoughts:  P  (  X  )  =      ∏          i      =      1        n              (      X      −              z        i            )                      m        i                  P    ′    (  X  )  =      ∑          i      =      1              n            m    i              (      X      −              z        i            )                      m        i            −      1            ∏          ?              ?                  (      X      −              z        i            )                      m        ?            I don&#039;t know with what i should fill the ?Then:                                                        P              ′                        P                                              =                                                            ∑                                  i                  =                  1                                                  n                                                            m                i                                                              (                  X                  −                                      z                    i                                    )                                                                      m                    i                                    −                  1                                                            ∏                                  ?                                                  ?                                                                              (                  X                  −                                      z                    i                                    )                                                                      m                    i                                                                                                      ∏                                  i                  =                  1                                n                                                              (                  X                  −                                      z                    i                                    )                                                                      m                    i                                                                                                                        =        ?            I&#039;m stuck here

gaiageoucm5p · Accepted Answer

You can act as follows:  P  (  X  )  =      ∏          i      =      1        n              (      X      −              z        i            )                      m        i              ⇒      P    ′    (  X  )  =      ∑    i    n        m    i    (  X  −      z    i        )                  m        i            −      1            ∏          j      ≠      i        (  X  −      z    j        )                  m        j              =      ∑    i    n        m    i    (  X  −      z    i        )                  m        i            −      1            ∏          j      ≠      i        (  X  −      z    j        )                  m        j              ⋅      (                  X        −                  z          i                            X        −                  z          i                      )    =      ∑    i    n        m    i    (  X  −      z    i        )                  m        i            −      2            ∏          i      =      1              n        (  X  −      z    i        )                  m        i            Hence                    P        ′            (      X      )              P      (      X      )        =      ∑          i      =      1        n        m    i    (  X  −      z    i        )                  m        i            −      2

Kendrick Hampton · Accepted Answer

The product rule generalizes such that the result is the sum of things where each thing is a derivative of one of the factors times the rest of the original products. For example   (  f  g  h      )    ′    =      f    ′    g  h  +  f      g    ′    h  +  f  g      h    ′  . So for your product, you want the product from   j  =  1 to   i  −  1 and then from   j  =  i  +  1 to   n

Let P be a polynomial of degree n , n ≥ 2 , P ( X

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