How to express the roots of a polynomial using its coefficients?

Question

Vieta's formula's can be used to express the coefficients of a polynomial using its zeros, but can they also be used for the opposite, by solving a system of equations described by the formulas? For example, if one of the coefficients contains a parameter, then how can I show what influence it has on the roots?

Answer 1

By the Abel-Ruffini impossibility theorem, there is no formula for degree five and above.

https://en.wikipedia.org/wiki/Abel-Ruffini_theorem

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If you want to know the influence of a single coefficient on the roots, say describe the relation between $x$ and the coefficient of the term of degree $k$ in

$$p(x)+\lambda x^k=0,$$

you can choose an arbitrary $x$ and deduce $\lambda(x)$. Then by differentiation,

$$p'(x)+\lambda'(x)x^k+k\lambda(x) x^{k-1}=0$$ gives you the sensitivity coefficient,

$$\frac{\lambda}{x}\frac{dx}{d\lambda}.$$