I'm a physical sciences grad student and I'm trying to model a system that contains 3 components. I'm not an expert in math, so apologies in advance if I say something stupid.
My problem boils down to solving a polynomial of the the form $ax^m + bx^2 + x -c = 0$ where $a,b,c$ are real positive numbers and $m$ is a very large even number ($~ 100$). Based on my experimental set-up, each composition is defined uniquely so there can only be one real positive root of this equation for a given set of $a,b,\text{ and }c$. For smaller m values (say 20), this is true and I actually do get only one real positive root but it doesn't quite work for larger m. I'm not sure if it's an artifact of the root finding algorithm I'm using or something else.
Is it possible to prove analytically that this equation will only have only one real positive root? I have tried applying Strum's theorem but I wasn't able to get a general solution from it. I can individually apply it on every polynomial but is there a way to show that any polynomial of this form will have one real positive root?
Thank you.
The answer is yes. This is a standard IVT+Role application, we often use problems like this in Calculus I.
Let $f(x)= ax^m + bx^2 + x -c $.
Since $f(0)=-c <0$ and $\lim_{x \to \infty}= \infty$ by the intermediate value theorem your polynomial has at leats one positive real root.
Now, since $f'(x)=max^{m-1}+2bx+1 >0$ for all $x \geq 0$ by Rolle's theorem your polynomial has at most one positive real root.
Therefore it has exactly one positive real root.
Note The claim fails if $a,b,c$ are not positive.