how to determine the nature of roots of a polynomial equation with degree higher than 30?

Question

the question says $P(x)=x^{32} -x^{25} +x^{18} -x^{11} +x^4 -x^3 +1$.how many possible imaginery and real roots does $p(x)=0$ has. how to determine the nature of roots for such equations of higher roots? note:By nature I mean no of possible real or imaginery roots.

Answer 1

Regardless of the degree, you can figure out how many roots, at maximum, are positive or negative. How? Decartes' rule of signs.

Here's how to apply it: The maximum number of positive roots of a polynomial $P(x)$ is equal to the number of sign changes in the coefficients of $P(x)$. The maximum number of negative roots is counted similarly in $P(-x)$.

Answer 2

$P(x) \ge 1$ for $x \le 0$ and for $x \ge 1$.

For $x \in [0,1]$, we have $P(x) \approx x^4−x^3+1$, which has no real zeros.

Therefore, it seems that $P$ has no real zeros.

This is not a rigorous argument though.