How the Roots of a Polynomial Depend on Its Constant Coefficient

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This Demonstration shows how the roots (blue points) of the polynomial depend on the constant coefficient , which is shown enclosed in parentheses. The polynomial has multiple roots (cyan points) if and . The values of for roots of the second equation are called critical points (red points).


The zeros of a polynomial are continuous functions of its coefficients. If , one root is 0, and the others are the roots of 1. In the case of , the roots are . If moves in a loop from to , enclosing only the positive critical point (red point), the positions of the zeros corresponding to and of as a function of are interchanged.


Contributed by: Izidor Hafner (May 2017)
Open content licensed under CC BY-NC-SA



This Demonstration visualizes a part of the proof that equations of degree 5 are not solvable in radicals [1, pp. 77–90].


[1] D. Fuchs and S. Tabachnikov, Mathematical Omnibus: Thirty Lectures on Classic Mathematics, Providence: American Mathematical Society, 2007.

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