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Suppose I have a polynomial equation of degree n, is there a way to find out if the equation has at least one real solution (not imaginary)?

And suppose a real solution exists, is there a way to find it out? (I know that Newton's method) can be used to approximate, but are there other methods)

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  • $\begingroup$ Are you asking for other numerical methods? $\endgroup$ Mar 23 '16 at 11:22
  • $\begingroup$ I guess you are thinking of a polynomial with real coefficients, not complex, but you might want to specify that... $\endgroup$
    – user193810
    Mar 23 '16 at 12:35
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To count the number of real roots, you can use e.g. Sturm's method (see my answer to Conditions for a unique root of a fifth degree polynomial). As far as I know, from a theoretical point of view, counting the number of roots is as hard as verifying that there is at least one (unless, as tilper correctly stated, the degree is odd).

To actually find the root, Newton's method can be a good choice if your input is "well-behaved" and/or you have a reasonable guess for the starting value. Otherwise, you should rely on classical polynomial root isolation methods such as algorithms based on Descartes' Rule of Signs. I mentioned some of them in Find all roots of a polynomial using secant method (which mainly refers to complex root isolation). You can find good (and free-of-charge) implementations in SAGE's real_roots package or the RS library by Fabrice Rouillier et al (disclaimer: I became one of the "al" recently).

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If the polynomial has odd degree, there will always be at least one real root. This can be seen with the intermediate value theorem.

If the polynomial has even degree then you can try precalculus techniques like the rational root theorem, synthetic division, Descartes' rule of signs, etc. Or you can analyze the polynomial and its derivative using Rolle's theorem and the intermediate value theorem, for example.

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