I took 3 random polynomials with non zero roots one having even degree and two having odd degrees
- f(x)=4x2−(4√3+12)x+12√3 having roots 3,√3 and leading coefficient 4 and calculated values of xf′(x)(f′(x) is the derivative of f(x)) at both roots which are 3f′(3) and √3f′(√3) and then sum of their reciprocals 13f′(3)+1√3f′(√3)=−112√3=−14(13⋅√3) then repeated same thing for
g(x)=1x3−203x2−12x+323 having roots 8,−2,23
18g′(8)+1−2g′(−2)+123g′(23)=11(18⋅−2⋅23)
h(x)=1x5+41x4+137x3−1601x2−1818x+3240 having roots 1,−2,5,−9,−36
11h′(1)+1−2h′(−2)+15h′(5)+1−9h′(−9)+1−36h′(−36)=11(11⋅−2⋅5⋅−9⋅−36)
Is this true for all polynomials? Is there any known result?
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