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1 year ago in Applied Mathematics , Mathematics By Govind

How can one show that a cubic equation has exactly one positive root?

As I’m modeling equilibria in a nonlinear system, the stability often hinges on the sign of the roots. I need a reliable, step-by-step analytical approach not just graphical intuition to prove uniqueness in the positive domain for parameterized cubics.

Equation CubicSpline SquareRoot CubeRoots

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By Kanishka Answered 1 year ago

In my work with nonlinear models, I’ve often needed to quickly ascertain the root structure of a cubic. I would recommend a combined approach. First, check the signs of the coefficients and the constant term to apply Descartes' Rule of Signs, which gives an upper bound. Then, evaluate the polynomial at zero and as it tends to infinity; if the signs differ, by the Intermediate Value Theorem, at least one positive root exists. To prove it's exactly one, examine the derivative. If the derivative is always positive (the cubic is strictly increasing), then that single sign change guarantees exactly one positive root. I've found this logic to be both rigorous and practical for parameter sweeps in simulations.

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