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I have a polynomial equation of $N$th order. The coefficients of the equation are parametrized by two variables, let's call them $a$ and $b$, both of which are real and positive. For general $N$, I can't write the solution as a function of $a$ and $b$ analytically, but it's ok because I don't really care what the solution is. What I do care is 1) does it have a (real) solution when the variable is in the interval [0,1]? and 2) if it does, how many solutions does it have?

Is it possible to answer these questions without actually solving the equation? If so, can you point me to relevant mathematical tools?

EDIT: Here is the exact problem I'm considering. I would like it to be more general, though, if possible because I might want to generalize my equation later:

$ (1-2x^2+x^4)(1+ax^N) = b^2 $

and I'm interested in the number of roots in the interval $x \in [0,1]$. If it helps, one can assume that $N$ is even.

Echows
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  • Does finding supremum/infimum of your polynomial on that compact help? If you plot this function for different $a$ and $N$, you "always" obtain that the $\inf=0$ and $\sup =1$. – TZakrevskiy Nov 19 '14 at 11:06
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    Can you get any way to the answer with Descartes rule of signs ? http://en.wikipedia.org/wiki/Descartes'_rule_of_signs – Tom Collinge Nov 19 '14 at 11:38

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