What will happen if there is a way predicting at a least one root of $p_{n}(x)=0$ without calculator?

Question

let $p_{n}(x)$ be a polynomial of degree $n$ defined as follow :

$p_{n}(x)=x^n +a_{n-1}x^{n-1}+a_{n-2}x^{n-2}+.....a_{0}$ which :

$a_{n-1},a_{n-2},.....,a_{0}$ are non nul real numbers coefficients.

my question is : What will happen if there is a way predicting at a least one root of $p_{n}(x)=0$ without calculator?

Any help is very welcom.Thank you