Why is sum of roots of quadratic equation power an integer coefficients of the expansion −f′xf(x)?Calculate the ∑1ann∈Zwhere a is the roots of a quadratic equation in the form f(x)=x2+ax+b.

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Corbin Pittman · Accepted Answer

Let f(x) be a monic polynomial of degree d with all roots&amp;nbsp;α1,…,αd&amp;nbsp;simple. In other words,f(x)=(x−α1)(x−α2)⋯(x−αd)=∏k=1d(x−αk).Then&amp;nbsp;ln⁡f(z)=∑k=1dln⁡(z−αk)and with&amp;nbsp;g(z)&amp;nbsp;=−f′(z)f(z), we observe-g(z)=f&#039;(z)f(z)=ddzlnf(z)=∑k=1dddzln(z-αk)=∑k=1d1z-αk.Now use the geometric series&amp;nbsp;1αk−z=∑n=0∞αkn+1znso that by comparing coefficients of equal powers of z in the power series, we conclude&amp;nbsp;an=∑k=1dαkn+1.

Why is sum of roots of quadratic equation

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