Thursday, January 17, 2019

calculus - Is this true for all polynomials

I took 3 random polynomials with non zero roots one having even degree and two having odd degrees


  1. f(x)=4x2(43+12)x+123 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)+13f(3)=1123=14(133) then repeated same thing for


  2. g(x)=1x3203x212x+323 having roots 8,2,23


    18g(8)+12g(2)+123g(23)=11(18223)





  3. h(x)=1x5+41x4+137x31601x21818x+3240 having roots 1,2,5,9,36


    11h(1)+12h(2)+15h(5)+19h(9)+136h(36)=11(1125936)



Is this true for all polynomials? Is there any known result?

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