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Let us consider the cubic equation

$$a(t)x³+b(t) x²+c (t)x+d(t)=0\qquad (*)$$

where $a(t),b(t),c (t),d(t)$ are rational functions defined in the whole real line minus some finite set of points. We know that (*) has still a real root $x(t)$ given in function of the coefficients (or in function of $t$).

My question is: How one can prove that the range (the codomain) of the real solution $x(t)$ is also the whole real line minus a finite set of points?

Maybe this link may help: Continuity of the roots of a polynomial in terms of its coefficients

Jean Marie
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Safwane
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    Is an inspection of the cubic formula, which gives an explicit solution for the equation in terms of $a(t),b(t),c(t)$ by radicals, not enough? I guess in particular it implies the theorem about the continuity of the roots in therms of the coefficients in the link you mention, in the case of degree 3 polynomials.

    http://mathworld.wolfram.com/CubicFormula.html

     – Victor Zhang Dec 16 '17 at 12:04
  • @VictorZhang: Can you elaborate with this – Safwane Dec 16 '17 at 16:24
  • There is a formula which gives you an explicit way to calculate the roots of a cubic polynomial equation in terms of the coefficients. It involves in principle easily understandable operations such as roots. I mean to suggest you to look at the formula, and see if it helps to answer the question. – Victor Zhang Dec 16 '17 at 21:27

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