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Consider a cubic polynomial of the form

$$f(x)=a_3x^3+a_2x^2+a_1x+a_0$$

where the coefficients are non-zero reals. Conditions for which this equation has three real simple roots are well-known. What conditions would guarantee that none of these roots is positive? In other words, what constraints on the parameters would guarantee that the polynomial has no positive roots? Please provide references also, if possible.

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    Evidently, the mathematician Sharaf al-Din Al-Muzaffar ibn Muhammad ibn Al-Muzaffar al-Tusi classified such cubics in his 12th century text Al-Mu'adalat (Treatise on Equations). In the treatise equations of degree at most three are divided into 25 different types. First al-Tusi discusses twelve types of equation of degree at most two. He then looks at eight types of cubic equation which always have a positive solution, then five types which may have no positive solution. Source: http://www-history.mcs.st-andrews.ac.uk/Biographies/Al-Tusi_Sharaf.html2012-09-16

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