I have this problem- lets say I have a polynomial which has real parameters as coefficients and I'm looking for the scope of the parameters where the polynomial can have real roots. e.g $x^2+kx+k$ we know that for $0<=k<=4$ we won't have real roots. Is there any algorithm for finding the scope of k for higher order polynomials? Thanks

edit: I think this can be found by Samuelson's inequality , solving what is in the square root will give the bounds. How can this be done to a polynomial with more than one varaiable?

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    $\begingroup$ Yes, there are many different ways of finding bounds on the roots of polynomials. When the algebraists proved the insolvability of general quintic and higher degree polynomials in the 1800's, the analysts came to the rescue inventing a number of ways of improving bounds on the roots. Some bounds are very rough estimates that are simple to apply, and others are quite delicate and crafty but more complicated to use. $\endgroup$
    – David H
    May 21, 2014 at 7:57
  • $\begingroup$ I know about bounds of roots. My question is can I know bounds on the parameter given ther is a root (doesn't matter wher is the location of the root) $\endgroup$
    – user1
    May 21, 2014 at 8:16
  • $\begingroup$ Maybe something there will help, if I can say that the bound on the roots is zero by Samuelson's inequality I can find bound on k $\endgroup$
    – user1
    May 21, 2014 at 8:20

1 Answer 1


I'm afraid one can't solve such a problem analytically for a generic polynomial even with the given coefficients. We have to resort to numerical methods. But then, we could consider the parameters as additional variables and solve the combined higher-dimensional problem, numerically, and possibly exploiting the fact that our new variables enter linearly.


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