Calculate $\sum^{20}_{k=1}\frac{1}{x_k-x_k^2}$ where $x_k$ are roots of $P(x)=x^{20}+x^{10}+x^5+2$

Since $$\frac{P'(x)}{P(x)} = \sum_{k=1}^{20}\frac{1}{x-x_k}$$

and $P'(x)= 20x^{19}+10x^9+5x^4$

we have $$\sum_{k=1}^{20}\frac{1}{1-x_k}=\frac{P'(1)}{P(1)} = {35\over 5}=7$$


Hint:

Set $y=1-x$. If the $x_k$ satisfy the equation $\;x^{20}+x^{10}+x^{5}+2=0$, the corresponding $\:y_k$ satisfy the equation $$(1-y)^{20}+(1-y)^{10}+(1-y)^{5}+2=0.$$

Can you find the constant term and the coefficient of $y$ in this equation, to use Vieta's relations?

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Polynomials

Roots

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