How to find real positive roots of high order polynomials with large coefficients? Can I scale it down while conserving the zeros?

Question

I'm trying to solve high order polynomials (~100) with really large coefficients. In my earlier post, I actually confirmed that these specific sets that I'm working with can only have one real positive root. But the root finding algorithm on MATLAB usually gives multiple real positive roots for orders >40 and larger coefficients. I was wondering if there is a way to scale down the polynomial with larger coefficients while conserving the real positive root? Thanks in advance!

Here's an example of such a dataset. http://www.filedropper.com/polynomials

Answer 1

Consider $p(x) = a_0 + a_1x +a_2 x^2 ....+a_nx^n$. If $\alpha$ is a root,

$$a_0+a_1\alpha +a_2 \alpha^2....+a_n \alpha^n =0 \\\implies \frac 1C (a_0 +a_1 \alpha +a_2 \alpha^2 ...+a_n \alpha^n)=0$$

You can make $C$ arbitrarily large to scale it down.