Show (if possible without simplifying) that f ( x ) = ∑ i = 1 n ∏ j ≠ i x − a j a i − a j = 1

Question

Show (if possible without simplifying) that   f  (  x  )  =      ∑          i      =      1        n        ∏          j      ≠      i                  x      −              a        j                            a        i            −              a        j              =  1

Mollie Roberts · Accepted Answer

Consider the degree   n  −  1 polynomial  f  (  x  )  =      ∑          i      =      1        n        ∏          j      ≠      i                  x      −              a        j                            a        i            −              a        j            We have  f  (      a    k    )  =      ∏          j      ≠      k                          a        k            −              a        j                            a        k            −              a        j              +      ∑          i      =      1            i      ≠      k        n        ∏          j      ≠      i                          a        k            −              a        j                            a        i            −              a        j              =  1  +  0  =  1The degree   n  −  1 polynomial takes the value 1 at n distinct values, hence it is the constant function 1Note that      ∑          i      =      1            i      ≠      k        n        ∏          j      ≠      i                          a        k            −              a        j                            a        i            −              a        j              =  0since one of the j will take value k

Show (if possible without simplifying) that f ( x ) = <munderover> ∑<!-- ∑ -

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